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~
l
LEt.! (mE)
Then if
k
( I I z. Ip) i=l
to complex valued polynomials of degree
Let
m denote a power of ~
II L II
mm
2.
)i¥i
m!
for any
L in t. s (mE) a
lip for aU
l
and
The next proposition gives, on taking
Proposition 1.11
Suppose
n +n +... +n = m. l 2 k
positive integers with
n l ,··· ,n k be are unit
A is the unit balL of E
p=2,
an extension of example 1.9
2n.
E denote a complex dlP
space,
l~p~=,
and Let
Then II LII
where the norms are taken over the unit baU of
E.
We conclude this section with two rather simple but useful results. The first follows readily from the binomial theorem and the second follows from an application of the maximum modulus theorem for functions of one complex variable. Lemma 1.12 in E and
L E ~
a
AE[
we have
(nE;F)
then for any
x,y
9
Polynomials on locally convex topological vector spaces
P (X+Ay)
and nl
P(x+y)
L
P(x) + P(y) +
(n) L(x)nr (y)r r
r=l Lemma 1.13
E and F are vector spaces over [, F, A is a balanced subset of E and
If
is a seminorm on
Moreover, if
AfO,AXEA
II PII S, x+A Proof
and
XEE
then
A is convex then
~ sup S(PCx+y» YEA
II pIIS,x+A
sup S(P(x+e yEA,SER sup s(P(e YEA SER
is
is
x+y»
sup S(P(y» yEA
y»
(since
A is balanced)
(by homogeneity)
(by the maximum modulus principle)
I~IIS, A· IfAxEA
then
x+A
C
lA +A A
§ 1.
2
CONTINUOUS
( 1 + l)A A
and hence
POL YNOI4IALS
Our main interest in this book is in studying continuous polynomial and holomorphic mappings between locally convex spaces.
However, we find
it useful (and necessary) to define more general classes of mappings this is analogous to the situation in functional analysis where the weak topology is used to obtain properties of the norm topology.
10
Chapter 1
Let [.
E and
F be topological spaces which are also vector spaces over
Usually there will be some relationship between the topological and the
vector space structures but for the moment we will not make any such assump
We let
tion.
J~s(nE;F)
and
nnE;F), J,.(nE;F)
denote respectively the
E into
spaces of continuous nhomogeneous polynomials from tinuous nlinear mappings from nlinear mappings from
E to
E into
spaces.
cs(E)
the con
F and the continuous symmetric
F.
In all cases we consider we find that
the above spaces are vector spaces and that A(J.. s(nE;F)) =y(nE;F).
F,
s ( j , (nE; F))
We now suppose that
E
and
'.[,s (n E ; F) and are locally convex
F
will denote the set of all continuous seminorms on
E.
If
is a seminorm on (E,a)/a 1 (0) Ea. Note that of
is
E a
E, Ea will denote the normed linear space will denote the canonical surjection from E onto a a(x) = a if and only i f II (x) = a and the open unit ball
and
II
a
lIa {XE:E;a(x) < I}. {xEE;a(x)
and
is the closure of mapping liS
0
f
Ba(r) in
{xEE;a(x):::r}. E.
Since
from a topological space into
is continuous for every
f
is a seminorm on
If
SEcs(F).
F
F
E
If
we let
aEcs(E)
lim
tEes (F)
then
(F ,II )' a S S
is continuous if and only if
We use this fact in extending
results concerning normed linear space valued polynomials and holomorphic functions to functions with values in an arbitrary locally convex space. Proposition 1.14
Let
E be a locally convex space over
normed linear space and suppose
n
P dP ( E;F). a
[,
F a
The following are equivalent
(aJ
P is everywhere continuous;
(bJ
P
is continuous at the origin;
(cJ
P
is bounded on some neighbourhood of the origin;
(dJ
P
is a locally bounded function (i.e. bounded on a neighbourhood of each point).
Proof 1.13
The implications (c) <=> (d).
suppose
A=P.
(a) => (b) => (c)
We now show
(c) => (a).
By the polarization formula and
are trivial and by lemma Let (c)
A E ~s(nE;F)
balanced neighbourhood of zero V such that IIAII Vn = M< Let be arbitrary. Choose a> a such that a Xo E V. By Lemma 1.12 00.
sup IIP(xo+liy)  P(x o ) II
YEV
:::
and
there exists a convex XOE E
11
Polynomials on locally convex topological vector spaces
I
n R=l n
:;;
(n) R
I R=l
(n) R
o
as
Hence
P
a
1 nR
M
I)
a nR I)
+
R
I)R
1 M[ (a + o)n _ ela ) n]
o.
is continuous at
Let
Corollary 1.15 P E ~a(nE;F).
let
sup II A(ax o ) nR (y) RII . yEV
x
and
o
and
E
(c) =>
This completes the proof.
be locally convex spaces over
F
P E ~(nE;F)
Then
(a).
if and only if
[
and
P is continuous at
one point. It suffices to use proposition 1.14 and the projective limit representation of
F by normed linear spaces.
We now look at a very useful factorization lemma.
If
E and
Fare
locally convex spaces,
a E cs (E) and P E @(n Ea ; F) then P 0 ITa E Jl(nE; F) may be identified with a subspace of ~(nE;F). When F
~(nEa;F)
Hence
is a normed linear space the factorization lemma says that the union of all ~(nE;F).
such subspaces covers
This is not surprising in view of lemma
1.13.
(Factorization Lemma).
Lemma 1.16
If
F is a normed linear space then
and
UaECS (E) for every positive integer Proof
P E ~(nE;F)
Let
.1"'>(nE
IT
a'
F)
n. A E ~s(nE;F)
and suppose
symmetric nlinear mapping.
Since
exists a e: cs(E)
Ilpll Ba(l) = M <00.
a(x) < 1, <1>[
I
E is a locally convex space
n
such that
a(y) = 0
and
EF'.
(~) A(x)R(Ay)nR] =
R=O polynomial from
c:
to
(;
I
F
is a normed linear space there
The function n
is the associated
Now suppose
gCA) =
(~)
R=O of degree (n.
X,YE E,
Since
is a
sup a(x+AY) = a(x) < 1 Ae:C
follows that
g
is a bounded polynomial and hence has degree
HahnBanach theorem the form
AX'
where
A(x)R(y)nR = 0 a(x')
for
O:SR:on1.
it follows that
o.
Since any
A(z)\y)nR = 0
it
By the ZEE if
has
Z e: E
12
and
.., P
Chapter 1
aCyl = 0, RFn. be defined on
Hence
P(z+y) = P(z)
by P(z) = P(Zl
E a
ITa (~l) = ITa. (z2) = z then ITa(z2zl) P(z2). Thus P is well defined and unit ball of lip
II
=
Ea
if
for any ze:E
satisfies
o and hence P 01Ta. = P.
if
z
aCyl =
o.
ITa(z) = z.
Let If
P(zl) = P(zl+(Z2zl)) =
Since
ITa (B (1)) a
is the
we have
II Pll Ba (1)
= M
<
00
and so
This completes the proof. The following example shows that this result does not extend to arbitrary
F.
Example 1.17 Let ~N denote the set of all sequences of complex numbers. N C is given the product topology or the topology of coordinate convergence and is a Frechet space (i.e. a complete metrizable locally convex space). N If a £ cs(C ) then E is a finite dimensional normed linear space and a n hence is isomorphic to C for some nEN. We claim
f:.
\.) :t(([N) ; CN) . ae:cs([N) a
belongs to ~~CN;CN) and I([N) = CN. If N L e: J(((CN)a;C ) then L((~N)) is a finite dimensional subspace aF.cs(C N) N aN of IN. Hence I i U ;i((C) ;C). a£cs(C N) a The identity
I
tJ
Our next result is a factorization lemma for arbitrary polynomials. The proof follows directly from lemma 1.16. Lemma 1.18
If
E and
n
BECS(F)
F are locally convex spaces over
(
then
\.) J)(nE;F ) aEcs(E) Cl B
Before defining hypocontinuous and Silva continuous polynomials we first prove a result about polynomials on a Banach space.
This result is true
for any Baire space and is used in proving Zorn's theorem in the next chapter. Lemma 1.19
Let
E be a complex Banach space and
locally convex space.
If
{Pm}:=l
F an arbitrary and is a sequence in
13
Polynomials on locally convex topolOgical vector spaces
fore1Jery
Pm(x)+P(x) Proof
in
x
E
thenp£@(nE;F).
We may suppose without loss of generality that
linear space.
Am £ ~s(nE;F)
Let
Am = Pm
where
By the polarization formula
Am
converges pointwise on
nlinear form
P
A
induction.
Let
B
is a normed
E
m.
to a symmetric
n
.~
A£.ra(E;F). We complete the proof by denote the unit ball of E. If n=l then
and by the uniform boundedness principle
Pm £ Y...(E;F) Hence
A and hence
F
for every integer
II pil B ~
M
result holds for
and n=k.
= M <"'. sUHlp m mliB is continuous by proposition 1.14. Now suppose the
P
If
Pm £ o>(k+lE;F)
for all
m then, by induction,
the mappings
are continuous for any
sup Yi £ B 2~ i~k+ 1
IIA
Yl' ... 'Yk+l £ E.
(Y2'··· 'Yk+l)
(x)
II
<
for every
'"
x
in
E.
By the uniform boundedness principle the collection of linear mappings is bounded on the unit ball of i.e.
E,
sup y.£B 1.
Id(k+l Thus
P
and
A are continuous.
This completes the proof.
The hypocontinuous and the Mackey continuous polynomial mappings between two locally convex spaces
E
and
F
either by defining new topologies on compact and bounded subsets of
An element
E.
P of ~a(nE;F)
can be defined in two different ways, E or by using properties of the
We describe both methods.
is said to be hypocontinuous if
Chapter 1
14
it is continuous on the compact subsets of
E.
We let
CPHyCnE;F)
denote
the vector space of all hypocontinuous nhomogeneous polynomials from E to F. ~aCE;F)
An element of
is said to be Mackey (or Silva) continuous
E onto bounded subsets of F. We let denote the vector space of all Mackey continuous nhomogeneous
if it maps bounded subsets of Q)MCnE;F)
E into
polynomials from
F.
Hypocontinuous and Mackey continuous nlinear forms are defined in an analogous way.
Since the vector sum of compact Crespo bounded) subsets of
a locally convex space is compact Crespo bounded) it follows from the symmetrization formula and the polarization formula that for any locally convex spaces
E and
F we have
SCJ.HyCnE;F))
J::.s
s (kMCnE;F))
:J M ( E;F)
HY(
n
E;F)
.s n
'*MCnE;F) " .x~CnE;F) and
A
(.t..'~(nE;F))
= (})MCnE;F).
X is a topological space we let X denote the"space X with the k finest topology which coincides with the original topology on the compact
If
x.
X = Xk we call X a kspace. If E is a locally convex space then it is not true in general that Ek is also a locally convex
subsets of space.
If
Any metrizable space is a kspace.
semi~lontel
barrelled
A locally convex space is a
space if its closed bounded sets are compact.
semi~lontel
space is called a Montel space.
An infra
The strong dual of
an infinite dimensional Frechet Montel space (a ~~lT) space) is an example of a kspace which is not metrizable.
The following proposition follows
easily from the definition above. Proposition 1.20
If
E and
F are locally convex spaces then
n
(j> HY( E;F)
Hence if convex space
E = Ek
then
(j) HyCnE;F)
(j>(nE;F)
fOT
any locally
F.
We now look at Mackey continuous polynomials. Let E be a locally convex space with topology T. A sequence CXn)n in E is said to be M x Mackey convergent to x, we write x >if there exists a sequence of n ' scalars, (An)~=l' I A 1+ as n+oo., such that A nCXnX) + 0 in (E,T) n
15
Polynomials on locally convex topological vector spaces
as
A subset
n + co.
M
as
+ x
A of
implies
n~,
E is said to be
XEA.
McZ.osed if
The Mclosed subsets of
(xn)n
E
£
A,
satisfy the
closure axioms for a topology which we call the topology of the Mclosure and denote by If
T
T . M TM
=
We also write
then the space
EM
in place of
(E,T)
(E,T M).
is called a superinductive space.
Fr~chet spaces and the strong duals of Fr~chetSchwartz spaces
are superinductive spaces. convex.
The topology
For instance if
(E,T)
TM
(~)d3 spaces)
is not necessarily locally
is the strong dual of a FrechetMontel
space then the following are equivalent; (a)
(E,T)
(b)
T
(c)
is a £H on
=
TM (E, T ) M
space,
E,
is a locally convex space.
Since closed sets are sequentially closed and Mackey convergent sequences are convergent Tk in
on any locally convex space
TM ;: Tk ;: T
is the ktopology associated with T. M (E,T ) i f and only i f x + x as n M n
Let
Proposition 1.21
any integer Proof
First suppose 13
£
cs(F)
E.
P
m
in
as
(E, T) .
Then for
£
13(P(xm))
i/213(P(Xm))
m for all
>
m2
>
m.
This implies that
I
for all
m.
This contradicts the
m
fact that x
M
m
1/2n
+
o
as
m+
oo
(since
m
x
1/4n (~ ) 1/2n m m
Hence
P
x ~
ml/4n
+
0
as
is bounded on bounded sets and
Now suppose
P
£
n+ oo
rP(n(E ) ;F) but P is not bounded on the M Then we can find a bounded sequence in E, (xm)m'
such that
l72n))
00
xn + x
be Zocally convex spaces.
F
x
13(P(
+
where
(i)(n(EM);F).
(ilM( E;F)
n
bounded subsets of and
E and
n
Note also that
(E, T) ,
(l>M(nE;F).
To show that
P E iP(n(EM);F)
it suffices
Chapter 1
16 M
P(xm) + PCx) as m+ 00 Let (Am)m be a sequence of scalars such that IAlm + + 00 and "m(XmX) + 0 as m + 00 . The set B {Am(XmX)}m u {x} is a bounded subset of E. If A E J:.sa (nE '·F) and A=P then A is bounded on Bn. Hence to prove that xm + x
as
m +00 implies
u
O:;:R::::n m = 1,2, ..
m=
is a bounded subset of
1,2, ..
F. R
It now follows that
nR
A(x x) (x) ... 0 as m'" 00 if 1:::: R::;n. Since m 'I n n R nR P(x m)  P(x) P(x+xmx)P(x) L R=l (R) A(xmx) (x) we have shown that P(x ) +P(x) as m ... '" and this completes the proof. m Corollary 1.22 (p a (nE; F)
for ever'Y
If
E and
F are locally convex spaces then
.;) tp M(nE; F) ::>
(j) HY (nE; F)
:.:) (? CnE; F)
n EN.
may also be regarded as a space of continuous polynomials if we E the finitely open topology t · A subset V of E , a vector' f is open for every finite dimensional subspace space, is t f open if V"F F of E where each finite dimensional space is given its unique norm
(f:l a (nE; F)
place on
topology. When do we have equality in corollary 1.22?
This is an interesting
question and the answer plays an important role in Chapter S.
We have
already noted cases in which we do have equality. These resulted from considering locally convex spaces as objects in the category of topological spaces and continuous mappings.
This category is much too large for our
purposes and rarely provides necessary and sufficient criteria.
In the
present case it provides sufficient but not necessary conditions for certain equalities.
On the other hand, we can also look at locally convex
spaces as objects in the category of topological vector spaces.
This cat
egory is too small and generally only gives conditions of the opposite kind  in the present situation we get conditions which are necessary. For example, in our terminology a locally convex space space i f and only i f
otM(E;F)
o((E;F)
E
is a bornological
for every locally convex space
17
Polynomials on locally convex topological vector spaces
F.
It is, however, useful to keep these type
of results in mind as they
provide useful estimates and give us our preliminary examples. The correct setting for the present chapter is the "category" of locally convex spaces and continuous multilinear mappings and we now give some examples which reflect the Example 1.23 and
FS
Let
E = F x
nature of this "category".
FS
where
is its strong dual (i.e.
linear
mappings on
bounded subsets of Let
A: E x E
is a locally convex space
is the space of all continuous
F with the topology of uniform convergence on the F). be defined by
+ [
1
A((x,x'), (y,y'))
2(x' (y) + y' (x)).
x'(x). a normed linear space. we always have
F'
F
A is continuous if and only if
E is
By the definition of the strong topology on
AE ~M (2E)
and frequently
A is
F'
hypocontinuous.
Our most useful counterexample in this book, [N x [(N), has the above form. and
eN [(N)
is the space of all complex sequences with the product topology is the space of all finite complex sequences with the direct
sum topology.
eN x e(N);
In later chapters we shall use the following properties of
it is a reflexive nuclear space with an absolute basis, it is
a strict inductive limit of Frechet nuclear spaces and an open compact surjective limit of ~~
(strong dual of Frechet nuclear) spaces, each
compact set is contained in a set of the form subset of
[N
and
KxL
where
K is a compact
L is a closed bounded finite dimensional subset of
[(N), every neighbourhood of the origin contains a neighbourhood of the origin which has the. form U x [N ~ x V where ~EN, U is a neighbourhood of zero in (~ and V is a neighbourhood of zero in [(N). Example 1.24
Let
E be a countable inductive limit of normed linear
spaces in the category of locally convex spaces and continuous linear mappings.
Thus
E
lim En
and
E
is a bornological space which con
+
n
tains a fundamental sequence of bounded sets, each
Bn
is convex and balanced.
(Bn):=l'
We may suppose that
Since a locally convex space is bornol
ogical if and only if every convex balanced set which absorbs set is a neighbourhood of zero, we find that sets of the form
ev~ry
bounded
\Ln=l An Bn
Chapter 1
18
form a basis of neighbourhoods of zero in all sequences of positive real numbers
E as
CL~=lAnBn
CAn)~=l =
ranges over
{L~=lAnbn;
bnE&n'
m
A B is convex and balanced and arbitrary}). This follows since ,00 Ln=l n n absorbs every bounded set and hence is a neighbourhood of zero. Conversely if
V is a convex balanced neighbourhood of zero then for every ntN 00 an there exists a > 0 such that aBc V and hence V ~ L 1 n B . n n n n 2 n The countable direct sum of normed linear spaces and the strong dual of a Fr~chet Montel space are examples of such spaces.
It can also be
easily shown that this class of spaces coincides with the class of bornological
DF
spaces.
We now show that for such convex space
F.
~)MCnE;F) = GlcnE;F)
for every locally
We may assume without loss of generality that
normed linear space.
A
E
P E ~MCnE;F)
Let
and suppose
F
A E~(nE;F)
= P. Since
Bl is bounded II p II Bl < M < have been chosen so that
A2 ,···, Am
m
II pll m ~ LA. B. i=l 1. 1.
M.
C2.
i=l
Let
00
1
and suppose
1 2i  1 )'
m
If x E 2.
i=l
A. B., 1.
and
Y E Bm+l
A> 0
then
1.
P(x)
p (x+>"y)
+
I
n
(~)A(X)nR(y)R. >..R
and hence
R=l II pll
Im
i=l
A. B. 1.
1.
where
Since
AE
o£
s n
~MC
E;F),
IIAI~
sufficiently small so that
is finite and we can choose
A
is a where
19
Polynomials on locally convex topological vector spaces
II P II
m+l
M(L
m+l
L
A.B.
i=l
1.
1
'1) .
i=l
21.
1.
By induction
we can choose a sequence of positive numbers
(Am):=l
such
that
II P II Hence
P
~
,00
is bounded on a neighbourhood of zero and is continuous by
proposition 1.14.
We also note that the above proof shows that if ~(nE;F),
is a subset of
(Pa)a£A
s~p
2M.
LAB m=l m m
II Pall Em
F
is a normed linear space, and
for every m then the collection bounded family of functions. < 00
Example 1.25
Let
convex space.
and only if each If each
E
J>(nE; F)
Em
is a metrizable locally
for every integer
is a normed linear space then
m
is a locally
n~ 2
if
is a normed linear space.
E m
E
by example 1. 24. Since
where each
Then
(P a)a£A
Conversely, suppose
El
(p
(nE; F)
is not a normed linear space.
is a countable inductive limit, it has a fundamental neighbourhood
system at the origin consisting of sets of the form
where Vm is m m and each compact subset of E
a neighbourhood of zero in
L:=lV
E for each mk is contained and compact in 'Lm=l Em for some positive integer
k.
Let
denote a sequence in (E)1 and let ~m F 0 £ E~ for every m~2. m=2 1 Using the natural embedding of each E m in E and the polarization formula Loo nl If (n E). Since it follows that P m=2<1>m~m £ a (",)00 ~m
k
PI
L
k
L
E m=l m
it follows that Now suppose
P
~nll
m=2 m m
k
L
m=l
~Hy(nE).
£
P
E m
£
~(nE).
neighbourhood of zero in
V m
Then there exists a sequence
Em' such that
V
m
a
20
Chapter 1
For each
m~2
choose
Ym
E
P(x+ym) = ~m(x)(~m(Ym))n1
Vm
~m(Ym)
such that
V < for every m~ 2. However if 1 able then it contains a neighbourhood
in
VI
then
such that
F
and
0,
such that Ei P such that P
for every integer Since
a:
m.
can be embedded
as a closed complemented subspace this ?(nE;F) f
example can be modified to show that is metrizable and not normable and
~Hy(nE;F)
whenever
EI
n~2.
We now introduce a further space of polynomials nomials.
~
is metrizable and not norm
00
in any locally convex space
x
<
Hence there exists a neighbourhood of zero
(~m):=2 a sequence in Hence we can construct
If
and hence I
I ~ mII
f O.
the nuclear poly
In contrast to the other spaces of polynomials we have introduced,
these form a subspace of the space of continuous polynomials. of nuclear polynomial allows us to regard p(nE) develop a duality theory.
The concept
as a dual space and to
The concept is also useful in developing the
theory of holomorphic functions on nuclear spaces. Denni tion l. 26 (a)
L
E
Let
E and
F be locally convex spaces.
.:t(n E ; F) is called a nuclear
n
linear mapping from
E
F if there exist a convex balanced zero neighbourhood U in E, a bounded subset B of F, i=l, ... ,n (Ak)~=l E £1 and sequences and i and (Yk)~=l where ~. k E
into
1,
k
and
Yk
E
B for all
k
such that
L(xl,···,x n ) L:=IAk~l,k(xI) ... ¢n,k(xn)Yk for all (xl' ... ,x n ) E En. We let i'NCnE;F) from (b)
E
into P
£
denote the space of all nuclear n
linear mappings
F.
(j>(nE;F) is called a nuclear
n
homogeneous polynomial
if there exist a convex balanced zero neighbourhood a bounded subset B of F, (Ak)~=l E £ I (~k) ~=l !:~ Uo and (Yk) k=l S B such that
U in
and sequences
E,
21
Polynomials on locally convex topological vector spaces
for every
P (x)
We let 0 (n E;F)
Taking
in
n=l
E.
in
denote the space of all nuclear nhomogeneous poly
N E into
nomials from
x
F. 1.26(a)
we obtain the definition of a nuclear linear
mapping between locally convex spaces. to be nuclear if for every
a E: cs (E)
that the canonical mapping from
Ef3
A locally convex space there exists
onto
Ea
f3 E: cs (E), f3,w,
is nuclear.
E
convex space
Proof A
in
I~=llanl I~n(x) I for every
~
~l
and
(~n)n
is said to be dual nuclear if If
Theorem I. 27
(An)~=l p(x)
such
This is equi
valent to the condition that for every continuous seminorm exists a sequence of E' such that
E is said
p
on
E
there
an equicontinuous subset
ES
x in E. is nuclear.
E is a nuclear locally convex space and
nE:N
A locally
then
The second equality follows immediately from the first since
(:iN (nE)) =
PN(nE).
L E: ~(nE)
show that if Since
By definition £N (nE) C vt(nE)
L
then
and so it suffices to
L E: iN(nE).
is continuous there exists
a E: cs (E)
such that
IL(xl,···,xn ) I ~ a(x l ) ... a(x n ) for any xl"" ,xn E: E. Hence, by the factorization lemma, we may look upon L as an element of t.(nE a ). Since E is nuclear there exists a f3 in cs(E) such that the canonical mapping from
Ef3
onto
(Yk)kCEa'
Ea
is nuclear.
(~k)k=ICE'
M<
Hence there exist
suchthat and
"',
{xE:E;f3(x)
<
U.
22
Chapter 1
LOOk
k lA ... Ak ¢k (xl) 1'···' n k 1 n 1
Since
and
is an equicontinuous subset of
E'
this completes the proof.
§1.3
TOPOLOGIES
ON
SPACES
OF
POLYNOMIALS
We now look at topologies on the various spaces of polynomials we have defined in the previous section.
Since \j)(lE)
=
E'
(algebraically), we
have various guides from the duality theory of locally convex spaces.
The
most useful topology from the linear viewpoint is the strong topology that is the topology of uniform convergence on the bounded subsets of
E.
This topology will be denoted by B. We shall see that it is not the most useful topology from the holomorphic point of view. and motivation.
However, it does serve a purpose in our development
For the sake of efficiency we shall always try to define
our topology on as large a space as possible. Definition 1.28 topology or the
Let
E and
F be locally convex spaces.
B topology on ~MCnE;F)
is defined to be the topology of
uniform convergence on the bounded subsets of l~M(nE;F),B)
by the seminorms
The strong
E.
is a locally convex space and its topology is generated IIC'!, B where
II
the bounded subsets of
ct
ranges over
cs CF)
and
B ranges over
E.
The following are easily proved. Proposition l.29 space, then
If
(PCnE;F),B)
Proposition 1.30
If
locally convex space then
E is a normed linear space and
F is a Banach
is a Banach space. E is a locally convex space and
F
is a complete
23
Polynomials on locally convex topological vector spaces
C1MC nE ;F), ~
is a complete locally convex space.
On ~HyCnE;F)
we naturally consider the compact open topology.
Let
Definition 1.31
E and
open topology on J)HyCnE;F) compact subsets of E. C~HyCnE;F)"O)
The compact
is the topology of uniform convergence on the
We denote this topology by
'0.
is a locally convex space and its topology is generated
II II Cl, K
by the seminorms
F be locally convex spaces.
over the compact subsets of
where
Cl
ranges over
cs CF)
and
K ranges
E.
Since every compact subset of a locally convex space is bounded it follows that S
='0
S ~
'0
on ?HyCnE;F).
By the HahnBanach theorem we have
if and only if the closed convex hull of each bounded subset of
is compact
i.e. if and only if
E
E is a semiMontel space.
Since the uniform limit of continuous functions on a compact set is continuous the following is true.
Let E be a locally convex space and F a complete locally conVex space. Then C@HyCnE;F).,o) is a complete locally convex space. By restriction Sand '0 define locally convex topologies on ~CnE;F). However, we shall need a further topology on
~roposition
~CnE;F).
define on
1.32
This topology, denoted by ~CnE;F).
'w'
is the strongest topology we
It is motivated by the factorization formula, the
definition of the strong topology on ~CnE;F)
when
E
is a normed linear
space and certain properties of analytic functionals in several complex variables theory.
It is perhaps more useful than
'0
or S
since it has
stronger topological properties but it is more difficult to characterize in a concrete fashion. We first consider polynomials with values in a normed linear space.
Let E be a locally convex space and let F be a Definition 1.33 normed linear space. The , topology on @(nE;F) is defined as the w
inductive limit topology in the category of locally convex spaces and continuous linear mappings ofO'(nE ;F), Cl
Cl
Cl E
cs(E),
lim CPCnE ;F),S). E'" cs CE) Cl
that is
24
Chapter 1
Hence a seminoY'171 every neighbourhood
of zero in
V
pep) : :
on J! (n E; F)
p
cCV)
E
i s , wcontinuous if and only if for there exists
such that
c(V) > 0
Ilpll v
We will subsequently see that this amounts to saying that a seminorm on
~(nE)
is 'wcontinuous if and only if it is ported by the origin and
for this reason we call
'w
( iP (nEa; F) ,8)
Since
the ported topology. is always a normed linear space
is a bornological space when Banach space
J,(nEa;F)
F
((}' (nE; F) "w)
is a normed linear space.
is a Banach space and
When
((jl (nE;F), 'w)
tive limit of Banach spaces, i.e. an ultrabornological space. ular,
(~(nE;F)"w)
F
is a
is an inducIn partic
is then a barrelled locally convex space, that is
every closed convex balanced absorbing subset is a neighbourhood of zero. For arbitrary
F we use the weak form of the factorization lemma , on
(lemma 1.18) and definition 1.33 to define Definition 1.34 defined on
Let
0(nE;F)
E
and
w
F be locally convex spaces.
Then
T
w
as
lim
lim 4
YECS
a
(F)
ECS
(E)
lim YEcs (F) The following elementary result shows the relationship between the topologies we have defined. Proposi tion 1. 35
For arbitrary locally conVex spaces n we have
any positive integer (a)
(b)
,
8
~,
8 and
'0
w
~
0
and
on define the same bounded subsets of
and hence have the same associated bornological topology.
E
~(nE;F)
F
and
is
25
Polynomials on locally convex topological vector spaces
We now give a number of elementary examples relating the above topologies  further examples appear in later chapters.
Afterwards, we define a
topology on the space of nuclear polynomials. Example 1.36
If
E
is an infinite dimensional Banach space and
a locally convex space, then T
W
=B~
Tw
on
TO
have the same bounded sets and hence
associated with
TO·
ExamEle 1.37
If
E
Banach space then the
T
T
(p (nE; F).
Moreover,
F T
o
is and
is the bornological topology
w
is a metrizable locally convex space and F is a bounded subsets of J,(nE;F) are locally bounded.
0
T B on Hence T is the bornological topology associated with T 0 w w IE) = E' if and only i f E is distinguished. Consequently i f E is a
a) (
nondistinguished Frechet space then Let
Example 1. 38 spaces and let
F
Since
T
TO'S
and
(s)
Let
Example 1.39
(nE) ,B)
Pn:E
subsets of
+
If a;N
Using the method of example
is metrizable and hence bornological. Moreover, if Tw
E = a;N x a; (N) .
S
E
TO
a;
be defined by a; (N),
Kl
and
TW
do not
E
and let
then there exist compact
K ' such that K c. Kl x K . Now every Z Z compact subset Of ((N) is finite dimensional and hence iiPniiK = 0 for all n sufficiently large. Thus B is a bounded subset of ((? (2E) ,TO). Let
and
TO
and hence are not equal.
Pn((xn)n'(Yn)n) = xnYn'
K is a compact subset of
There
is semiMontel and hence a on ~\(nE;F).
We show that
define the same bounded subsets of J'(2E) Let
E'.
define the same bounded subsets of
TW
space, it follows that
B = (Pn);=l.
on
E be a countable inductive limit of normed linear
B on rJ' (nE;F).
w
TO
E contains a countable fundamental system of bounded sets
it follows that fore
S
be a normed linear space.
1.24 we see that
'Y (nE; F).
f f
TW
and
N
un
(0, ... ,1,0, ... ) E a; and let ~ ntil position eN)
(P, ... 'k ,0, . .. ) E I[ . Let ~ nth position defined by
p
denote the semi
pep)
If
a
E
cs(E)
then
a(un,O)
o for all n sufficiently large. Hence
Chapter 1
26
= PCO,vn) for all n sufficiently large and pCP) < 00 for every 2 P in Ij) C E). Since the seminorm which maps P E 9C 2E) to I PCnun,v n ) P((O,vn)1 i s , 0 and hence 'w continuous and ((f (2E) "w) is barrelled it follows that p is a 'w continuous seminorm on :YC 2 E). Since p(Pn)=n PCnun,v n )
for all
n
we have shown that
We shall see later that
B
is not a ,
(:r (2E), '0)
~ (2E) .
bounded subset of
w
is a bornological space and
the barrelled topology associated with
is
~C2E).
on
'0
'w
This result extends easily to ~(nE).
P NCnE),
We now define three topologies on correspond to
and
'0,(3
'w
lIo' lIS
and
If
respectively on iJlCnE).
convex sUbset of a locally convex space
and
E
L
E
lIw. B
is a balanced we let
tNCnE)
~k.E E'}
and each
,
~
where the infima are taken over all possible representations of lIBeL) lIB(L)
and
Crespo P) hood
and
V
lIBep)
TIBep) E
may be infinite.
are always finite.
ikN(nE)
(respectively
of zero such that
TIvCL) <
These
However, if
B
Land
is bounded then
Moreover, by definition, if
!Y NenE))
1
then there exists a neighbour
and
lIv(P) <
topology on
j:. (nE)
00
p.
These allow us to
00
give the following definition.
(a)
Definition 1.40
The
TIo
the locally convex topology generated by subsets of (b)
lIK
(resp.
N
is
K ranges over all compact
as
E.
the
lIS
topology on J:N(nE)
(respectively ON (nE) ) is the
locally convex topology generated by
TIB
as
a
ranges OVer
all bounded subsets of E. (c)
A seminorm be
TI
w
p
on £NCnE)
(respectively ~N(nE))
continuous if for every neighbourhood
V
is said to of zero
in
27
Polynomials on locally convex topological vector spaces
there exists
E
c(v)
>
such that
0
(respectively
peL) ( c(v)lIV(L)
is the topology generated by aU
lIw
Since subset J.N(nE)
IILllBn (lIB(L)
B of
E
for all
Let
Proposition 1.41
balanced compact subset
lI o )
for every convex balanced
lis) Sand
TO'
lIw)
TW
on
n.
E be a quasicomplete dual nuclear space.
on ~N(nE)
TO
continuous seminorms.
lIw
IlpII B( lIB(P)
it follows that
and f N(nE)
lIo = liS = S =
and
pcP) ( c(V)lIV(P))
K
for every of
E
Then
Moreover, for each convex
n.
there exist
cK> 0
and a convex balan
ced subset Kl of E such that \ (L) ( c~ II L II (Kl)n and for every nonnegative n.
for every
L in
~N(nE)
Proof
Since
E
is quasicomplete and dual nuclear its closed bounded
sets are compact and hence
E'
Since set in
S
E,
S =
TO
and
is nuclear we can, given
lIo = liS. K
a convex balanced compact sub
choose a convex balanced compact subset
such that the canonical mapping from E~l
>
E~
Kl of
is nuclear.
p, k) ~=l E Q, l' (ak ) ~=l a sequence in Kl (since ive) and a bounded sequence (~)oo in E' such that "'k k=l Kl
exist
for every
Now let a A
E
in E' Kl L
(£N(nE),lI O ), in.t N(nE).
E
E
is semireft'ex
such that
a (L)
K.
There exists, by the HahnBanach theorem,
Now suppose
K.
containing K
Hence there
where the series converges uniformly on
J:N(nE) .
verges uniformly on
E
a(L) = lIK(L)
and
L r:=l ~m ,
1···
We then have
for every ~
m,n
where the series con
28
Chapter 1
This completes the proof. Combining theorem 1.27 and proposition 1.41 we obtain the following result. Corollary 1.42 then
If
:J:. (nE) = oi:N(nE)
Theorem 1.43
(t(nE),T) o Proof
If
and
ITo =
for every
TO
n.
E is a quasicomplete dual nuclear space then
is a nuclear space. If
for every
E is a quasicomplete nuclear and dual nuclear space
~
K is a compact subset of
in
E'.
Hence if
n
E then there exists a compact
is any positive integer and
then sup Yi E K l:oi:onl
II LII n K
(
sup Yi E K l:oi:onl
sup IL(y l ,· .. ,Ynl,Y) I yEK
rm=l
I a mL(yl,···,y n 1,x)1 m
(and by induction)
L
E
~(nE)
29
Polynomials on locally convex topological vector spaces
Since a compact sequence
this completes the proof.
The form of the above inequality will be used in chapter 3.
Let
Proposition 1.44 ,
IT
w
on ~(nE)
w
E be a quasicomplete nuclear space.
for any positive integer
Moreover, for any convex balanced neighbourhood exist a convex balanced neighbourhood such that
c>o
Proof IT , w w
IT w
~
W of zero, contained in V, and Lin £C n E),n=1,2, ...
'w it suffices to prove the above inequality to show
By the nuclearity of bourhood
E we can choose, given a convex balanced neigh
V of zero, a neighbourhood of zero
ical mapping
Ew
EV
7
is nuclear.
in
L
E
of (nE)
and
such that the canon.
IILII n
(An)n
x = 2~=lAk4>k(x)xk
E
Now suppose
We V
Hence there exist
WO and (xk)~=l c V such that E where the convergence is in EV'
(4)k)~=l
V of zero, there
for every
ITwCL):scnIILllvn
Since
Then
n.
E
t ,
l for all
1.
V
and hence L
Now
IL(x
k1
II L II n :s 1 V
, ... ,x
kn
) I :s 1
for any choice of Hence
<
x
k
, ... ,x 1
kn
in
V
since
x
Chapter 1
30
Let
c
then
t\11
Thus if
llW(L)
>;
IILII Vn
=
L\
vn
L
and hence
II Vn
) >; c n .
IILII Vn
c n II L II Vn and since this inequality is trivially satisfied we have completed the proof.
00
The preceeding results can be transferred to
n
homogeneous polynom
ials by using the inequality
L in X s (nE) n ~N (nE) = ~~ (nE)
for any B of
E.
In particular, we find
and any convex balanced subset
(~N(nE),rro) ~ (~N(nE),rro)
and
(:J,. ~(nE) ,llw) ~ ((j) N(nE) ,llw).
We now summarize results obtained in this way. Proposition 1.45 let
n
Let
E be a quasicomplete locally convex space and
be a nonnegative integer. (aJ
If and
(bJ
II i3 = 0 " is a nuclear space.
E is dual nuclear then C;(nE), TO)
T
0
on a:>.\l (nE)
E is nuclear then 'J' (nE) = (j' (nE) N and II w T on J' (nE) w
If
Moreover, the estimates given in propositions 1.41 and 1.44 are still valid, with minor modifications, for spaces of homogeneous polynomials on the appropriate locally convex spaces.
31
Polynomials on locally convex topological vector spaces
§1.4
DUALITY THEORY FOR SPACES OF POLYNOMIALS In this section we consider linear functionals on the locally convex
spaces of polynomials defined in §1.3.
This topic is currently the subject
of research and should play an important role in the development of the subject in the near future.
Our presentation of results is not fully com
prehensive but hopefully outlines the main developments and provides a glimpse of future developments. We show that continuous linear functionals on spaces of polynomials can themselves be represented by polynomials.
Our main tool in obtaining
this representation is the Borel transform.
Let
Definition 1.46
A be a vector spac? of [
polynomials defined on a locally convex space form on ".
The Borel transform of T, BT,
valued nhomogeneous
and let
E
is defined on
T
be a linear
{
by the formula BTC
is a subspace of E* and AOS CFn)c\4 then the restriction of is an nhomogeneous polynomial. For example i f fJ, = (j) CnE) a J:> CIlE) then BT e:;P CnE*) and if ~~ or \?N CnE ) then BT e: ? CnE'). a a In the cases we consider Pr is a locally convex space and T is continIf
BT
to
uous.
F
The Borel transform will only be useful if it is injective.
will always be the case i f 1\ property and 1'1
=
=
PNCnE)
or if
This
E has the approximation
J) CnE) .
Proposition 1.47 The Borel transform is a vector space isomorphism from nE (i) U NC ),I1 ), onto IfJCnES) and (ii) C(j)N CnE ),I1 0 ), onto ?CnCE~To)) B Under this isomorphism the equicontinuous subsets of (~N(nE),I1B)' corr
espond to the locally bounded subsets of subsets of ~CnCE,To)')·
Proof
(f CnE ),I1 ), o N
If> (nEs)
and the equicontinuous
correspond to the locally bounded subsets of
Since both cases are proved in a similar fashion, we only consid
er the case There exist
(ONCE) ,liB). c>O and
B is linear and injective. Let T e: CQlNCnE) ,I1r)' B an absolutely convex bounded subset of E
such that IT(p)
I
~
c
II
BCP)
for every
P
in CP NCnE) .
32
Chapter 1
In particular if
and
11<1> liB" 1
then
IBT(
= IT(n)1 "
c II
(,:' N(nE) ,liS)' P' E I? (nES')
convex bounded subset and let
B of
and II p'll E.
If
?N(nE), i.e. the definition of
the representation of
P.
Moreover, if
for some closed absolutely
"c
BO
P E :P N(nE)
,'" II <jl nil < '" for some neighbourhood V L i=l i V T(P) = I:=lP'(i)' One easily shows that
ear operator on
:p (nES)'
is a locally bounded subset of
,00
n
and P = L i=l
is a well defined lin
T(P)
is independent of
II
then
P'(
and
hence I T(P) I
"
inf.{I:=ll P' (i)1 ;
P
r
. 1<1>·n1 } 1=
r
n II B . I P' ( inf.{ i=lll
) I; P
r
n i=l
II i II B (
inf .{
Hence
L:=lll
c
and
BT(
liB; P =
L:=l
= T(n) = P' (<1».
is surjective and a vector space isomorphism.
This shows that
B
The above also shows that
for any
c>O
and this completes
the proof.
The Borel transform is a vector space isomorphism from
Proposition 1.48
C? N(nE) ,II() ,
onto fi; (nE ,)
(the space of n
homogeneous polynomials on
E' which are bounded on the equicontinuous subsets of E'). Under this isomorphism the equicontinuous subsets of (~jN(nE),IIw)' correspond to subsets of ~i;(nEB) subsets of Proof
which are uniformLy bounded on the equicontinuous
E'.
The proof is very similar to the proof of proposition 1.47.
T E (~N(nE)'IIw)'
and let
V
Let
denote an absolutely convex neighbourhood of
33
Polynomials on locally convex topological vector spaces
o
in
E.
There exists
II BT II VO
Hence
c(V»
0
such that
I BT (~) I
sup ~EE '
11~11~1 and
BT
£
?~(nE')'
sup
11~11~1
IT (~ n) I : :
c (V)
This also shows that the image by
B of an equicontin
(5' (nE) IT )' is a subset of ? (nE,) consisting of funN ' w ~ ctions which are uniformly bounded on the equicontinuous subsets of E'.
uous subset of Now suppose
P'
£ ,/ ~
(nE').
We define
is a neighbourhood of zero in
Moreover, since isomorphism.
BT(~)
E and
= P'(~)
T
as in proposition 1. 47.
PEP (nE) N
If
V
then
this shows that
B is a vector space
The result about equicontinuous sets also follows from the
above. Quite a number of corollaries can be deduced from the above. just a few examples.
The first is perhaps the most interesting.
We give Since
the collection of spaces which occur in this corollary is rather interesting, (see chapters 3 and 5), we give them a special name. Definition 1.49
ES
A LocaLLy convex space
E is fuLLy nucLear if E and
are both compLete infrabarreLLed nucLear spaces. A fully nuclear space is a reflexive nuclear space and the strong dual
of fully nuclear space is fully nuclear.
Every Frechet nuclear space is
fully nuclear. Corollary 1.50 If E is a fuLLy nucLear space then '0 = 'won n a positive integer, if and onLy if J> (nE,) = (PeTIE') and the
6> (n E), B
ID
bounded subsets of ~M(
n ,
E B)
M
are LocaLLy bounded.
B
B
34
Chapter 1
Proof
Since
E
is an infrabarrelled locally convex space the equicon
tinuous subsets of ~ CnE') ?cC nE '). M
S
E6
coincide with the bounded sets and hence
It now suffices to apply theorem 1.27, and propositions
"
1.45,1.47 and 1.48 to complete the proof. In particular we note that nuclear space.
'0
Also this shows that
,w on YCnE) if E is a , t T w on :?Cnq;CN) x 'IN)
Frechet if
0
n:::2,
a result which we have already proved directly Cexample 1.39). Corollary 1.51
If
E is a reflexive nuclear space then
Corollary 1.52
If
E is an infrabarrelled locally convex space then
Corollary 1.53
If
E is an infrabarrelled
Frechet space then
TIS
TIw
DF space or a distinguished
on ?NCnE)
We now look at some examples in which the Borel transform gives a topological isomorphism.
a locally convex space, has property
E,
subset
We first need some preliminary results.
K
of E
E such that
(s)
if for each compact
there exists an absolutely convex bounded subset
K is contained and compact in
B
of
E . B
(E is the vector subspace of E generated by B and endowed with B the norm whose unit ball is B). If B is complete then EB is a Banach space.
Strict inductive limits of Frechet spaces and strong duals of
infrabarrelled Schwartz spaces have property fully nuclear space has property Lemma 1.54
= JlM(nE;F)
Proof
If
In particular, every
(s).
If the locally convex space
~Hy(nE;F)
(s).
(E,,)
has property
for any locally convex space
F and any
(s)
then
nsN.
is compact in E, then there exists an absolutely convex such that K is compact in EB· Hence T ,T M and II II B induce the same topology on K. If P s O"MCnE;F) then pl K is 'M and hence T continuous. Hence P E: (J'HyCnE;F) and (j)MCnE; F) = (JlHycnE;F). set
B in
Lemma 1.55
K
E
If
E
is a fully nuclear space then (J> HY (nE)
is equal to
35
Polynomials on locally convex topological vector spaces
the completion of Proof convex space
~) Hy(nE)
E and thus to prove this result it suffices to show
lies in the completion of
(5,(nE) ,TO).
K be an absolutely convex compact subset of
Let
E.
A
Since
€
t:.~y(nE)
and let
E is dual nuclear
and quasicomplete we can choose subset of
Kl , Kl<K, an absolutely convex compact E such that the canonical injection E + EK is nuclear.
Hence there exist (Yn):=l (~Kl
(An):=l
€
(CPn)~=l
.\',1'
such that for every x
x
in
K
€
Eic
we have for any
1
Ilnll K ~ 1
and
K
L~1= lA.CP.(x)y. 1 1 1 Since
where the series converges uniformly in
K~
with
xl, ... ,x n
A is continuous on
K
€
A(t1= lA..(xl)y·,···,t 1 1 1 1= lA.CP.(x l I n )y.) 1
A.
1
n
A(y. , ... ,y. ) 11 In
<1>.
11
(Xl)
Now
sup IA(y ...... y. ) I 11 In i l ,···, in choose a finite set of indices
F
<
00
Hence, for any
°
>0,
we can
such that
A. A(y. , ... ,yo )cpo •.• IIA  LFA. ¢i II K < 0/2. Since E' is dense 1n 11 11 lk 11 k in E' K we can choose a sequence of continuous linear forms on E, (l/Ji)7=1' such that < 0/2 l/J II L cP II  L l/J i i i i F l n F l n K Combining these two in equali ties we obtain the desired result. If
(E,T)
iated with
T
is a locally convex space then the Mackey topology assoc(not to be confused with the topology of the Mclosure TM)
is the finest locally convex topology on dual as
(E,T).
E which has the same continuous
If the Mackey topology associated with
T then we say that
CE,T)
is a Mackey space.
T coincides with
An infrabarrelled locally
36
Chapter 1
convex space is a Hackey space.
CPHyc nE ) ,TO)
Proof
E is a fully nuclear space then
If
Proposition 1.56
is the completion of a nuclear space and hence it
is a complete nuclear space. CPCnE') ,: (:))
e
HY
CnE) T)'
Let
TS
be the strong topology on
Cproposition 1.47).
C:J'CnEJ),T
'0
"
is the strong
Q )
"
dual of a semireflexive space and hence is a barrelled Mackey space. (j)(nES),TW)
is also a barrelled space and hence a Mackey space.
To com
plete the proof we need only show
Since
CPHyCnE) ,TO)
osition 1.48,
is semireflexive,
C~CnES),Te)'
CG)CnES),T W), ,: O'MCnE) = :PHyCnE),
0Hy CnE).
=
By prop
and this completes the
proof.
If
Proposition 1.57
E
is a fully nuclear space then
CQ CnE) , Tw) i,
)J
\I
if and only if the
Tw

CJ;)HY (n EB' ,) T0 ) .
~CnE)
bounded subsets of
are locally bounded.
Proof If CPCnE),Tw)S': C(i)HyCnES),To) then by proposition 1.56 C(1PC n E),T )6)' ,: ~(nE) and hence C~CnE),T) is a barrelled semireflexive w "
w
space and thus it is reflexive.
Hence
C0HyCnES) ,TO)
is also reflexive
and the equicontinuous subsets of the dual coincide with the strongly bounded subsets.
The strong topology on
(~HyCnES),To)'
is the TW
top
ology by proposition 1.56 and the equicontinuous subsets are the locally bounded sets by proposition 1.47. of G'CnE)
Conversely if the
TW
are locally bounded then the bounded subsets of
bounded subsets (U>HyCnES) ,TO)'
are equcontinuous Cpropositions 1.47 and 1.56). Hence C~HyCnEB),To) is infrabarrelled and thus reflexive. By proposition 1.56, C~CnE),Tw) = ~YHyCnES),To)B
is also a reflexive space and
This completes the proof. We now look at linear functionals on spaces of homogeneous polynomials
Polynomials on locally convex topological vector spaces
defined on Frechet spaces with the approximation property.
developed and indeed the general theory for
Nuclear poly
We study only
nomials still appear and play an important role. uous forms as the corresponding theory for
37
T
o
contin
continuous forms is not yet
Tw TO
continuous forms is almost
an untilled field. One can easily show that the vector space isomorphisms of proposition 1.1 yield a topological isomorphism when the appropriate spaces are endowed with the compact open topology.
This imp11es the foll
owing result. Lemma 1.58
If
is a Frechet space and
E
then for any positive integer
Proposition 1.59
If
convex space and
T
~
\T(L) \ for all
L in
E
E
F
is a Frechet space,
(<x(E;F)
is a locally convex space
n
F
is an arbitrary locally
satisfies
,TO) ,
cl\Ll\y,K
~(E;F)
where
YEcs(F)
(xn)n
in
and
K is a compact subset of E
then there exist (a)
a null sequence
(b)
asequence
(~n)n
in
for all
in
and all
y
F
F'
E, suchthat
\~n(Y)\iiCY(Y)
n
and
Ln \A n \~l
(c)
such that
00
T(L)
In=lAn~n(L(xn))
for every Proof
L
in
~(E;F).
The compact subset K of E is contained in the absolutely conx >0 vex hull of a null sequence, i.e. there exists n as n
Hence
such that
38
Chapter 1
Thus
IT(L) I Since
L
c sup y(Lx ).
~
n
E ~(E;F),
Yn
E
n
(L(xn))n
Fy and
is a null sequence in
y (Y ) .... 0 n
as
n .... oo}.
c (F ) o y
F.
Let
c (F ) o
y
is a normed linear
space witn norm
By the HahnBanach theorem we can extend co(Fy)
II(U } ) I ~ c supy(f ) n n n n
Since
(¢n)~=l
T to a linear functional
T
on
such that
~l((F
c o (F y )' E
(F y) , C F'
y )')
where
for every
{f }
n n
00 00 Iit I <: 1 \' nn=l E ~ l'Ln=l 11 ' , l¢n(y)1 ~ cy(y) for all Y in F and all
there exist
(it)
n
such that
'T(U } ) n n
Hence for any
L
in
~(E;F)
we have
and this completes the proof. Proposition 1.60
Let
E be a Frechet space and
F
a locally convex
space. If T E (:t:(nE;F), TO)' satisfies IT(L) I ~ c IILI16,Kn for every L in X(nE;F) where K is an absolutely convex compact subset of E and 6
E
cs(F)
then there exist an absolutely convex compact set
~
K and:
39
Polynomials on locally convex topological vector spaces
(a)
a sequence in
(b)
a sequence for all y
(4)k):=1 in F' such that in F and all k,
such that
Proof T
L
in
We proceed by induction on
n.
Proposition 1.59 covers the case
Assume the proposition is true for the positive integer
s
it
l4>k(y)I li:cS(y)
T(L)
for every
n=l.
~, {xl , ... ,x }oo k nk k=l
£(n+lE;F).
where
Define
of lemma 1.58.
Ci.,
Iii II (S,K) ,Kn
T on
Ts
*(nE; £(E;F))
by the isomorphism, call
sup IIL(Yl'·"'Yn)11 (S K) Yi sK ' i=l, ... ,n
= sup S(L(YI'''''Yn)(Yn+~) Yi sK i=l, ... , n+l
a sequence in ~ where
{(xl, .. ·,X )}koo_l' k nk Ii:
cIIMlts,K)
for every
M in
(.t:.(E,F),,)' o
and
By proposition 1.59, there exists a null sequence in each
k
and for
there exists a sequence of scalars
l~k,j(Y)1
Ii:
cS(Y)
Let
(.t:(nE;(.;((E;F)"O))"O), and
Hence, by induction, there exist
l4>k(M)1
n.
for all
Y in
F and all
j
such that
40
Chapter 1
for every
M in
ot:: (E ;F) .
Hence T(L)
T(a(L))
f or every
L
'n ~rn+lE,·F). ~
l
we may reorder the above to obtain a sequence with the required properties. This completes the proof.
Proposition 1.61
If E is a Frechet space with the approximation prop
erty, then the BoreZ transform,
B, is a Zinear isomorphism from onto
Proof T
E
Since
(~(nE), TO) I
( (il N (n (E ', T
E has the approximation property
o
)),
IT ).
w
B is injective.
Let
and suppose
IT(P) I ~ where
K is an absolutely convex compact subset of
E.
By the polarization
formula and proposition 1.60 there exist a relatively compact sequence {xm}:=l and (Am):=l E ~l' I:=lIAml ~ c such that 00
T(P)
Lm=lAmP(xm)
Hence BT(~)
BT
T(~n)
n Lm=l Amxm
00
for all
n
Lm=lAm~ (xm)
00
E
P
in
(p
(nE) .
for all
(i'N(n(EI ,TO))'
~
in
E'
and
Polynomials on locally convex topological vector spaces
A further application of proposition 1.60 shows that (~CnE),,)' o
that the equicontinuous subsets of the form ~ CnE)
{p E if'NCnCE"'o));\\P\\KO ~ cKL = ~~CnE)
CE'"o)'= E and
closed convex hull of sets of the form
U{PE lPNCnCE'" KCE K compact
));\\p\\ 0
N,K
0
are a fundamental neighbourhood system at K
is surjective and
correspond to sets of
it follows, by proposition 1.48, that the
oCi5'NCnCE'"o)), (PCnE))
c
Since
B
41
~ cK}
0
in
CtP NCn(E "'o»:I1w)
as
ranges over all possible sets of positive numbers.
A fundamental neighbourhood system at by the polars of bounded sets. Since C
n
{p
E:
tPCnE);\\P\\K
0
in
C~CnE)"o)B
~
KCE
K compact
0((
U
[p (nE) " o )',!p (nE)) {PE
P (nE); \\P\\K ~
KCE
K compact
This completes the proof.
closed convex hull of cK}O
is given
42
Chapter 1
§l.S
EXERCISES
The following exercises develop topics which we shall encounter in later chapters and also certain material which we did not find convenient to include in the text. difficult.
Consequently, some of these exercises are rather
A serious attempt at solving them, will, however, provide a
good deal of insight into the theory fying nontrivial problems.
even if only as a means of identi
For the research worker they could easily lead
to new techniques and worthwhile research projects.
Starred exercises are
commented on in Appendix III. 1.62
~a(mE;F)
Show that
Dim(E) :s 1 1.63*
or If
ra (nE) 1.64
if and only if either
E,F
for all and
n
if and only if
E ~ [CN) .
G are vector spaces over
[,
P E cPaCE;F)
Q Ef'a(F;G)
show that
QoP E1='a(E;G).
1.65
E and
are locally convex spaces and
that
If
P
m=l,m=O,
is an infinite dimensional locally convex space, show that
E
= (p(nE) If
oC!(mE;F)
F = {a}.
F
is continuous at one point if and only if
P
and
P E (p a (E; F)
show
is everywhere
continuous. 1.66
Replace continuous by hypocontinuous (resp. Mackey continuous) in
exercise 1.6 S. 1.67
If
E
convex space and cp
0
P E 6>(E)
1.68*
Let
mapping from
is a metrizable locally convex space, P E 9 a (E;F)
for every E and E
into
cP
in
show that
P E ~(E;F)
Let
is a locally
F'.
F be real Banach spaces and let F.
F
if and only if
Llyf(x) = f(x+y)f(x)
f
be a continuous
for all
x,y
in
and define LlYl LlY2 ... LlYnf(x) inductively. Show that f is a polynomial of degree :Sn if and only if LlYl ... 6Yn+lf(x) = 0 for all Yl""'Yn+l and x in E. Show that this result does not extend to Banach spaces over the complex field.
E
43
Polynomials on locally convex topological vector spaces be a Fr~chet space and suppose P E: P a (nE). is continuous if its restriction to a 2nd category subset of
1.69*
Let
E
Show that E
P
is
continuous. 1.70*
i: functions
E = .J) = Space of
Let
be the Dirac delta function at the point E: (p (2o'V)
and that
(f ):=2 m Show that
(f!(2J)),TO).
1.71*
a.
I~=l(3noa).
1.72*
for all
If
show that
E
n
and
1.73*
Let
E = lim E
R.
We
E::J) I
(ano ).0 a
:r Cn J) I), T) o
n
= f
m
are locally
~y(n.i)l) =(f'(nJ)I). Conclude that t TW on (?C n 1)l) for all n32.
(9
is a basis for
C¢n)n
TO=TW
E'
is a basis for
S
(2 E),T ). O
be a strict inductive limit of Frechet Montel
m
>
m Show that the following are equivalent:
spaces. (a)
each
(b)
,~ (nES)
(c)
!p(nE') S
1.74
Let
(¢n)~=l _ ,00
TO
(
is a Frechet nuclear space and
(¢n¢m)n3m=1
m I n=l
Show that
°a
on E: :?Hy(2,'i))"!p(2;:)).
Show that the bounded subsets of
on (f(n.}))
Let
is a Cauchy sequence in
bounded and hence deduce that
E:
E~
E m =
r? Hy(n ES )
and let D \r
0
2
m:( E) V
(2E) ,TO)
n32,
for some for all
nE:N.
where each
Ern
!PHy(nEs ) = Lm=O Ern
a neighbourhood
(lP
admits a continuous norm,
,00
E
P  Ln=l ¢n~n E:
that
of compact support in
J) with its usual strict inductive limit topology.
endow
t ~n
E:
E~
and that
is a locally convex space.
for all P E:
of zero such that is not complete if
fP (I
n.
2 (E)
i f and only i f there exists
II
Let
Show that
00
for each
n.
Hence show
is a nonnormed metrizable
locally convex space. 1. 75 P
E:
x as
E:
,00 E where each E Let E Lm=l m m is a Banach space. Let (p (n E) and for each positive integer m let Pm(x+y) = P(x) where
,m Lj=l m >
,00
Ej
and
y
E:
Lj=m+l Ej .
Show that
Pm
E:
(fI (nE)
and that
00 uniformly on a neighbourhood of each point of E.
P
m
>
P
44
Chapter 1
1.76*
If
E
is a metrizable locally convex space and
integer, show that the compact open topology on locally convex topology on
~ (nE)
~ (nE)
If
E
~M(E;F)
tially continuous polynomials from Let
is the finest
cP (nE) .
is a locally convex space in which every null sequence is a
Mackey null sequence, show that
1.78
is a positive
which coincides with the topology of
pointwise convergence on every equicontinuous subset of 1.77
n
is the space of Fvalued sequen
E into
F.
A denote an uncountable set.
~A
Show that
is not a
kspace, but that for all 1.79
n.
Show that the following two conditions on a locally convex space E
are equivalent: (a)
every compact subset of (i)
(b)
E is strictly compact;
every null sequence in
E is a Mackey null
sequence; (ii)
every compact subset of
E
is contained in the
absolutely convex hull of a null sequence. 1.80
If
E
is a locally convex space, show that
kspace associated with 1. 81
E
If
in
E.
P
E
TO
fr> (n(ExF)) a
Let X
the space of topology.
F
is a Banach space, show that
bounded if and only if
00
E.
E and
If
1. 83*
(E',o(E',E)).
is a Frechet space and
is
true for arbitrary 1. 82*
is the
sup I P(x) I < for every PsB Construct a counterexample which shows that this result is not
B C~(nE;F) x
(E',T ) O
F
are both Frechet spaces or both JJ '1 Trz. spaces and
is separately continuous, show that
P
is continuous.
be a completely regular Hausdorffspace and let ,£ (X)
~valued
continuous functions on
Show that for each
n
X with the compact open
be
45
Polynomials on locally convex topological vector spaces
6'(n£,(X))
(a)
p (ni.,
(b)
1.84* P
E
P
E
If
E = Co(f),
(f> (nCo (f ) ; F)
onto
C (f ).) o 2
to its coordinates.
P
L . (L is the natural proj ection f f 2 2 Show that {~nl~ E Co(f)'} spans a dense C
x
L
E
E
2
and suppose
r r
[P(x)](t)
P 1.86*
'r
P
K E L2([0,1]n+l)
Let
for every
X is Lindelof.
(tP (nC o (r))'S).
subspace of
Let
if
uncountable,
such that
2
1.85
(X))
F = 2 (f l ), fl uncountable and show that there is a countable f2 in f and
~(nE;F)
f
X is paracompact;
if
o
o
([0,1]).
K is symmetric with respect
K(t l ,···, tn' t)x(t l )·· .x(tn)dt l ·· .dt n
Show that
'" n L2 [O,l];L 2 [0,1]).
\r (
E is a separable Hilbert space and
If
there exists an
x
in
E, Ilxll= 1
and
a
A in
PE
6' (nE;E)
(,
show that
IAI = 1,
such that AP(x)
1.87*
E and
If
11~lx.
=
F
are Banach spaces, we say that T
weakly compact if it maps the unit ball of compact subset of
F.
If
T
E
P (E;F)
E
P (E;F)
is
E onto a relatively weakly
show that the following are
equivalent (i)
T
is weakly compact,
(ii)
T*
E
(iii) 1. 88*
(f> (F';E')
T** (E") C
(the adjoint of
A Banach space
into
that
is weakly compact,
F. E is said to have the polynomial DunfordPettis
property if for every Banach space E
T)
F
the weakly compact polynomials from
F map weak Cauchy sequences onto strong Cauchy sequences.
Show
E has the polynomial DunfordPettis property if and only if every
Banach valued weakly compact linear mapping maps weak Cauchy sequences onto strong Cauchy sequences.
46
Chapter 1
1.89* p
E:
If
f> (F)
1.90*
F
is a nuclear subspace of a locally convex space
show that there exists If
E
.p
E:
is a Banach space, show that
such that
E and
pi F = P.
((fNCnE),ITw)
is also a
Banach space. Tf denote the topology on @aCnE) of uniform convergence on the finite dimensional compact subsets of the vector space E and let 1.91 *
Let
E~
(E*,crCE*,E)).
Show that the Borel transform is an algebraic isomor
cU' aCnE) , Tf)'
phism from
onto
cP CnE*cr)
Let (x ) ~=l be an orthonormal subset of a Hilbert space n m E: I? N( mE) m be a positive integer. Show that P n=l "nxn only if I:=ll"n l < and also that
i f and
1.93*
if and
1. 92*
r
let
E and
00
If
E
is a locally convex space, show that
only if for each locally convex space
I EJ.. where
we have
E' = F' 8
8
1.94
If
m and
n
1.95*
1.96*
E as a subspace
{¢EF';¢IE=o},
CdPCmE),T) o
m~n,
and
E
is a locally
is isomorphic to a complemented sub
CrPCnE),To)' A compact Hausdorff space
closed subset of that
E' = (E' , Tw) 8
which contains
are positive integers,
convex space, show that space of
.J.
E
F
X is said to be dispersed if every
X contains an isolated point.
e($' en J:, eX)) , S)
If
X is dispersed, show
has the approximation property.
Show that n times
§1.6
NOTES AND REMARKS Mathematicians began exploring the concepts of polynomial and holomor
phic mapping in infinite dimensions at a time when ideas and theories such as the total derivative, point set topology and normed linear space, etc.
47
Polynomials on locally convex topological vector spaces
were either still in their infancy or not yet discovered.
Moreover, it
appears that the search for fundamental concepts in infinite dimensional differential calculus stimulated much of the work which resulted in the satisfactory linear theory that we now know as functional analysis.
These
pioneers were motivated by many different considerations, and at times were not aware of one another's work.
We provide here a brief outline of the
early development of polynomials, a similar treatment of holomorphic functions is given in §2.6, and refer to the historical survey of A.E. Taylor [680] for further details. It is generally recognised that the definitive step in the creation of infinite dimensional analysis was taken by V. Volterra in 1887.
In a
series of notes [705,706,707,708,709], which later evolved into the book [710], he developed a theory of scalar valued differentiable functions on ,~[a,b]
and obtained the following Taylor expansion [70S,p.10S] for the
realvalued analytic function
y
on
.t. [a, b]
y 1[q,(x) + ljI(x)] 1
J~ .. J y(n)I[
yl[q,(x)]1 + r  l  J 1 II (n) a ~,lj!
where The
n
E
th
on ,(i;, [a,b].
.e,[a,b]. term in the above expansion is an
n
homogeneous polynomial
Volterra did not, however, specifically discuss polynomial
mappings. The next step was taken by D. Hilbert who outlined a theory of holomorphic functions in infinitely many variables at the international congress in Rome in 1908 and published his results the following year, [332].
To Hilbert, each variable was a coordinate evaluation and he used
a monomial expansion with absolute convergence on polydiscs as we do in chapter 5.
Each holomorphic function had, in his notation, the following
Taylor series expansion ~(Xl,x2'···) = c+
Lc x + L c x x + L c x x x + (p) p p (p,q) pq p q (p,q,r) pqr p q r
48
Chapter 1
the series converging
absolutely on
Ix31
:;: 1£ 3 1 , . . . . It is clear from the above that Hilbert had a definite concept of polynomial in infinitely many variables. During the same year, 1909, M. Frechet published his first contribution [240] to the abstract theory of polynomials in infinitely many variables.
Motivated by Cauchy's observation that any continuous real
valued function
f
of a real variable which satisfied the·equation
f(x+y)
f(x)  fey) = 0
had to have the form
for all
x,y
in
R
f(x) = Ax, he gave an abstract "difference" charact
erization of real polynomials of one or several real variables (see exercise 1.68).
He then used this characterization to define real polynomials
depending on a countably infinite number of variables. was
RN
His domain space
and on it he defined, for continuity purposes, a metric which
gives the usual coordinatewise convergence topology.
The ,following year
he used the same method in [241] to define real polynomials on )6 [a,b] and showed that a real nhomogeneous polynomial
U on this space could be
represented as Uf = lim ffi>
where
r a
ll~m)
is a sequence of nhomogeneous polynomials in
independent of ~
[a,b],
f
xl"" ,x n and the limit is uniform over the compact subsets of
He also showed that any polynomial could be represented as a
finite sum of homogeneous polynomials.
In a subsequent paper, [243), he
obtained a Riesz representation theorem for bilinear forms on The next step is due to R. G~teaux.
£
[a,b] .
He made very fundamental contrib
utions to the theory of infinite dimensional calculus (see §2.6), and his simple elegant style makes for very pleasant reading.
Gateaux's work
consists essentially of two papers [252,253], which he wrote during the period 19121914.
He died in 1914, and his results were edited by P. Levy
and published in 1919 and 1922. KN
(K=IR
or
Gateaux worked only on the spaces
HenotedthatFrechet'sdefinition 2 of polynomial was inadequate for functions defined on vector spaces over a:),
9.
ande,[a,b).
49
Polynomials on locally convex topological vector spaces
the field of complex numbers and proposed instead that a continuous function ~
P
such that
for any vectors
degree
n.
z
P(AZ+~t)
and
t
is a polynomial of degree
n
in
A and
in the domain be called a polynomial of
He showed that his definition coincides with Frechet's for real
valued functions of real variables and went on to prove various results such as the relationship between the homogeneous parts and the "Gateaux" derivatives of a polynomial  with his definition.
The development of the
concept of normed linear space and associated ideas between 1910 and 1925 allowed Frechet to extend his definition of real polynomial to a rather general setting in [244] and [246]. In 19311932, A.D. Michal, a student of Frechet, gave a series of lectures at the California Institute of Technology in which he outlined the relationship between symmetric nlinear forms and homogeneous polynomials. This relationship had been noticed earlier for bilinear forms and 2homogeneous polynomials by M. Frechet [243] and R. G~teaux [252]. Further work on the definition of polynomial between Banach spaces was carried out by A.D. Michal and his students A.H. Clifford, R.S. Martin, I.G. Highberg and A.E. Taylor [331,492,493,449,677].
R.S. Martin, in his thesis [449]
proved the polarization formula and I.G. Highber£ [331] clarified the relationship between the different definitions and showed that Frechet's difference method could be extended to the complex case if one added the hypothesis of
G~teaux
differentiability.
Independently, S. Mazur and
W. Orlicz [481,482], established the connection between the nlinear approach and the now classical approach of Fr~chet and G~teaux for real Banach spaces, and proved the polarization formula.
This ends our brief
sketch of the development of the concept of abstract polynomial.
For those
interested, we strongly recommend the original sources as interesting reading. We return now to commenting on the text and will try to attribute results to their original sources.
There are basically three approaches
to studying polynomials, by considering restrictions to finite dimensional spaces,by means of tensor products
and by using multilinear mappings.
of these methods are useful and none should be neglected.
All
The restriction
method, as already noted, was one of the original methods used and reappears in our work every so often.
The tensor product approach is due
to R. Schatten [626] and A. Grothendieck [287].
In [620], R.A. Ryan shows
Polynomials on locally convex topological vector spaces
50
that most of the results we present can be obtained by this method and the same approach is also to be found in C.P. Gupta [295,296],
T.A. Dwyer
[214,215,217,218,220,223], A.Co1ojoara [138,139] and P. Kr~e [402].
We
follow the multilinear approach in this book. Theorem 1.7 is due to R.S. Martin [449] and an alternative proof is given in L.A. Harris [310].
Example 1.8 is due to L. Nachbin [509] and
example 1.9 was discovered independently by O.D. Kellogg [379], J.G. van der Corput and G. Schaake [169] and S. Banach [45].
The proof given here
is due to S. ~ojasiewicz and can be found in [73].
Propositions 1.10 and
1.11 are due to L.A. Harris [310] and further similar results may also be found in that article
and in [316].
The factorization lemma (and the corresponding result for holomorphic functions) has been implicit in the works of many authors, e.g. A. Hirschowitz [335], C.E. Rickart [605] and L. Nachbin [514].
A system
atic study of this idea and its consequences is undertaken in S. Dineen [190] and E. Ligocka [443].
All results using surjectiye limits (see
chapter 6) depend in some way on a factorization property. due to S. l1azur and W. Orlicz [482].
It
Lemma 1.19 is
is also a consequence of the
uniform boundedness principle for polynomial mappings on a Banach space and more general results on the same topic can be found in J. Bochnak and J. Siciak [73], P. Lelong [431] and P. Turpin [687].
J. Bochnak and
J. Siciak [73,74] make extensive use of the "Polynomial lemma of Leja" [424] in proving their results. Proposition 1.21 is well known, and probably due to J. Sabastiao e Silva.
The result of example 1.24 is proved for a countable direct sum of
Banach spaces in S. Dineen [185] and for ~ 1lrLspaces in S. Dineen [194]. The proof given here is modeled on those given in [185] and [194] and related results are to be found in [50].
Example 1.25 is due to P.J.
Boland and S. Dineen [92] and the proof is similar to that of the particular case
q;N X a: (N)
which appears in S. Dineen [185].
The method of
proof has been further developed by L.A. de Moraes [498] in her study of holomorphic functions on strict inductive limits. Nuclear polynomials on Banach spaces were introduced by C.P. Gupta
51
Chapter 1
[295,296,297] in order to prove existence and approximation properties of convolution operators on Banach spaces and motivated L. Nachbin [508,509] to introduce the concept of holomorphy type.
This allows one to discuss
compact, integral, nuclear, Hilbert Schmidt, etc. polynomial and holomorphic mappings between locally convex spaces  concepts which have proved useful in linear functional analysis and in probability theory on locally convex spaces (see appendix I). P.J. Boland [79,82,83,84] initiated and developed the theory of holomorphic functions on nuclear spaces and for such spaces nuclear polynomials playa fundamental role (see chapters 3 and 5).
Recent work by
R.A. Ryan [620] has shown their importance for holomorphic functions on Frechet spaces with the approximation property. The strong topology and the compact open topology are derived from functional analysis and point set topology respectively.
The
TW
topol
ogy is more or less special to the theory of holomorphic functions on locally convex spaces, although some linear properties of this topology are discussed in R.E. Edwards [229, p.5ll5l3], K. Floret [237], G. Kothe [397, p.400] and J.A. Berezanskii [57].
This topology was introduced by
L. Nachbin [509] and was motivated by results of A. Martineau [450,453] on analytic functionals (of several complex variables) supported by every neighbourhood of a compact set but not by the compact set itself.
It may
also be described as the topology of local convergence. Example 1.38 is given in S. Dineen [185] for a countable direct sum of Banach spaces. sets of a
The same result for holomorphic functions on open sub
/JJJR space is proved by J .A. Barroso, M.e. Matos and L. Nachbin
[50] and for ;j)Jh1 spaces by S. Dineen [194].
Example 1.39 appears in
S. Dineen, [185]. In dealing with nuclear polynomials and the Borel transform we have attempted to correlate various results on nuclear and dual nuclear spaces by P.J. Boland, [79,82,83,85], results on fully nuclear spaces by P.J. Boland and S. Dineen [90] and results on Frechet spaces with the approximation property due to R.A. Ryan [620].
This leads to a more compact
treatment but essentially the only new results in the final two sections are propositions 1.44 and 1.48.
Definition 1.40(a) is due to P.J. Boland
52
Polynomials on locally convex topological vector spaces
(79].
The fundamental inequality needed in definition 1.40(b) is also due
to Boland [87].
The definition is, however, new although special cases
have previously been considered by C.P. Gupta [295] and R.A. Ryan [620]. Proposition 1.41, corollary 1.42 and theorem 1.43 are due to P.J. Boland [79,83].
Proposition 1.44 is proved for fully nuclear spaces in [87].
The Borel (or FourierBorel) transform was first used in infinite dimensional holomorphy by C.P. Gupta [295].
Subsequently, it has been
applied by various authors, see chapters 3,5,6 and appendix I, in the study of convolution operators and duality theory.
Proposition 1.47 is
due to C.P. Gupta [295] for Banach spaces and to P.J. Boland [79] for semireflexive nuclear and dual nuclear spaces.
Other particular cases of
proposition 1.47 and 1.48 can be found in R.A. Ryan [620].
Fully nuclear
spaces (definition 1.49) were defined by P.J. Boland and S. Dineen [90] and figure prominently in chapters 5 and 6.
Corollary 1.50 and lemma 1.55
are proved for fully nuclear spaces with a basis in P.J. Boland and S. Dineen [90] and the latter result has recently been extended to quasicomplete dual nuclear spaces by J.F. Colombeau, R. Meise and B. Perrot [153].
Lemma 1.54 is due to S. Dineen [190].
Propositions 1.56 and 1.57 are due to P.J. Boland and S. Dineen [90] when
E has a basis, and to P.J. Boland [87] in the general case.
proof given here of proposition 1.57 is new. 1.59, 1.60 and 1.61 are all due to R.A. Ryan.
The
Lemma 1.58 and propositions They appeared in a prelim
inary draft of his thesis [620], but were replaced by more elegant and perhaps slightly less general results in the final version.
Chapter 2
HOLOMORPHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES
In this chapter, we give the various definitions of holomorphic mappings between locally convex spaces, which we shall use as well as the different topologies on these spaces of mappings.
In many cases, it is only
necessary to consider Banach space valued mappings and the results may be extended to arbitrary vector valued mappings quite easily. The compact open topology and the
'w topology easily extend from
polynomials to holomorphic functions on locally convex spaces, but the strong topology does not generalise in a suitable fashion.
While the
'w
topology plays an important role in our study, it does not, in general, have good topological properties.
We introduce the
'0
topology which may be
described as the topology supported by the countable open covers.
We prove
elementary properties of these topologies and give some simple examples .. The significance of these examples and counterexamples is clarified in later chapters.
The remaining chapters are devoted to a deeper study of
these topologies on certain classes of locally convex spaces and on special domains.
We also define germs of holomorphic functions.
These are of
intrinsic interest and also playa role in duality theory.
§2.l
1\
GATEAUX
Definition 2.1 open if U (I F
HOLOMORPHIC A subset
~~PPINGS
U of a vector space
dimensionaZ subspace
tf.
F
for each finite
F of E.
The finitely open subsets of ogy
E is said to be finiteZy
is an open subset of the Euclidean space
The balanced
neighbourhoods of zero.
tf
E define a translation invariant topol
neighbourhoods of zero form a basis for the
On a locally convex space
(E,,),
than any of the topologies we have previously considered on
53
is finer E,
i.e.
tf
Chapter 2
54
A function
Definition 2.2
defined on a finiteLy open subset
E with vaLues in a Locally convex space
a vector space Gtlteaux or
f
Gholomorphic if for each
a£ U,
F is said to be
and q, £ F'
b £E
U of
the complex
vaLued function of one complex variable A + q,of(a+Ab) is holomorphic in some neighbourhood of zero. set of all
We let
U into
GhoLomorphic mappings from
HG(U;F)
denote the
F.
Hartog's theorem in finite dimensions says that separately holomorphic U>
functions on
is Gholomorphic i f and only i f q, of
Iu f"lG
is a holomorphic function of
several complex variables for each
q,
F'
subspace
G of
E.
in
and each finite dimensional
Consequently, one can use any of the equivalent
finite dimensional conditions (e.g. Taylor series expansions, Cauchy Riemann equations, existence of the total derivative) in the definition of Gholomorphicity. At this stage, one may wonder why we demanded a locally convex range space in definition 2.2. vector space over
a:
We could also give an analagous definition with a
as the range space.
However, we would then run into
difficulties in showing that a Gateaux holomorphic function has a Taylor series expansion about each point since this requires a convergence structure on the range space and the finite open topology on very few functions would have the desired expansion. convex range space we still have to be careful.
F
is so fine that
Even with a locally
We first show that Gholo
morphic functions are continuous for the finite open topology.
If
Lemma 2.3 F
E is a vector space,
is a locally convex space and
f
E
U is a finitely open subset of
HG (U; F)
then f
E,
is continuous when
U is given the finite open topology. Proof
It is easily seen that t f is the inductive limit topology, in the category of topological spaces, given by the inclusion mappings G~E
where
G ranges over all finite dimensional subspaces of
function
f
defined on a
tf
open subset
U of
E.
only if its restriction to the finite dimensional sections of uous.
Hence a
E is continuous if and U are contin
Since an analytic function of several complex variables is continuous
this completes the proof. We now look at Taylor series expansions of G~teaux holomorphic functions.
5~
Holomorphic mappings between locally convex spaces
If
Proposition 2.4 F
U is a finitely open subset of a vector space
is a loca Uy convex space and
f
E
then for each
HG (U; F)
exists a unique sequence of homogeneous polynomials from
t;
E
U
E,
there
E into
/'\
F,
}'" { 'dmf~t;) m. m=O
such that
f(t;+y)
in some
y
for aU Proof
Let
XEE
t
t;EU
f
neighbourhood of zero.
be fixed.
For each positive integer
m,
E F'
and
let Pm, t; , (x)
of(i;+Ax) Am+l
J
trri
dA
i"i=p x px
is chosen so that
t;+{Ax;i"i~p }CU.
Pm, t; , (x)
is independent of
Px
where
f
f
is Gholomorphic
and I ZITi
of(t;+x) By lemma 2.3,
Since
x
Df(t;+Ax) dA. "m+l
is continuous and we may use Riemann's definition of the
integral to define f(t;+Ax) "m+l
I
m,,,~ (x)
P
ZITi
J iAi=p
dA.
x
The limit (of the Riemann sums) may not exist in in
f\
F,
the completion of
F.
It will lie in
F
F but will always exist if
F
is sequent iall y
complete or i f the closed convex hull of each compact subset of F is compact. For this reason, we sometimes place a completeness condition on the range space. all
and
Since
f
is continuous,
Pffi,s,,+, ~ '" (x)
<j>(P
m,,,~ (x) )
for
x.
Since the restriction of
of
to any finite dimensional section of
U
is a holomorphic function of several complex variables, it follows that the function
x
and each
mapping on
+
in E.
P
~ ",(x)
m,s,,'t'
F'.
Let
lies in ~ (mE) a
L m, E;,
for every positive integer
be the associated symmetric mlinear
m
56
If
Chapter 2
xl, ... ,x m E E then by the polarization formula (theorem 1.5) 1
Lill,s,,+, C" '" (xl' ... , x ) m
I~.=±l El···E mPm,e,,'!' C" ",(I~ 1 E.X.) ~ ~= ~ ~
m!
1
m!
Let
L
m,E;
: Em
F
>
be defined by 1
Lm,sC" (xl' ... ,x m)
m!
By the HahnBanach theorem Let
1. as ( mE; F)
and
Pm,.,e
V be a tfbalanced neighbourhood of zero such that
example, take
V
{xEE;E;+AXEU,iAi"l}).
a compact subset of E;+{h;iAi"p} CU. y E F}.
LeE ill,s
XEV
If
U and hence there exists 6Ecs(F)
If
By lemma 2.3
f
let
p>l
then
(for
E;+{AX;iAi"I}
is
such that
{q,EF';iCd~6(y)
B6
E;+V(:U
for all
is continuous and we have
sup 6(f(E;+Ax)) iAi"p
sup iq,of(E;+AX) i i Ai"p,q,EB 6
M6,x < '" .
Hence M
~(P
,,(x))
sup jp "",(x)i q,EB m,e,,'!' 6
m,e,
for every positive integer
m.
,,~ p
m
This shows that
Hence f(E;+x)
I:=o
Pm,E;(x)
for every
x
in
V.
Using the uniqueness of Taylor series expansions in one complex variable we
57
Holomorphic mappings between locally convex spaces
see that the sequence
(Pm,~):=o
is uniquely determined by
The finite dimensional theory also shows that partial derivative of
f
at
in the direction
~
f.
Pm,,,~(x) x,
is the
th
m
and following
classical terminology, we write P
The corresponding
'clmf(O m,~
m!
m linear form
Our expansion now becomes
L
is denoted by
m,~
m!
"m
I:=o d f(~) (x). m!
f(~+x)
proof.
This completes the
In proving proposition 2.4, we have also shown the following: Proposition 2.5 (Cauchy inequalities)
and in
cs(F)
and every nonnegative integer
II
§2.2
If
B is a balanced subset of E such that
;!
HOLOMORPHIC
Definition 2.6
1 m P
MAPPINGS
Let
E
a finitely open subset of
BETWEEN and
E.
F
m!
~E U
O
PEnt, {O}
then for every
S
sup
S(f(x))
xE~+pB
CONVEX
SPACES
be locally convex spaces and let
A function
"m ,00 d f(O Lm=O
~+pBCU
m
LOCALLY
if it is Gholomorphic and for each y
f E HG(U;FL
~
in
f:U U
U be is called holomorphic the function
~ F
( )
y
converges and defines a continuous function on some ,neighbourhood of zero. We let H(U;F) U into F.
denote the vector space of all holomorphic functions from
We usually consider functions defined on open subsets of
E
and in
this case, because of the uniqueness of the Taylor series expansion and the fact that the finite open neighbourhoods of zero are absorbing, a Gholomor
Chapter 2
58
phic function is holomorphic if and only if it is continuous. The following observation is easily proved and frequently applied.
If
Lemma 2.7
U is an open subset of a locally convex space
a locally convex space and TIaof
E
for every
H(U;F a )
A continuous function
f
f
then
HG(U;F)
E
in
a
f
E
f
if and only if
cs(F).
with values in a normed linear space is
locally bounded (i.e. each point in the domain of whose image under
H(U;F)
F is
E,
is bounded).
f
has a neighbourhood
The converse is false in general but it
is true for Gholomorphic functions as our next result shows.
If
Lemma 2.8
U is an open subset of a locally convex space
a normed linear space and
then
f E HG(U;F)
f
E
H(U;F)
E,
F
if and only if
is f
is locally bounded. Proof
Let
~
E
U be arbitrary.
hood of zero such that
~+V
By proposition 2.5, for all
Choose f(~+V)
C U and
m and
0
<
3
a. o
as a + then we can choose Hence we have
II f(~ a )
 f(~)
is a bounded subset of
0 < I,
00
a
V a convex balanced neighbour
a
o
F.
we have
such that
for all
II 00
Im=l
\1
~d f(~)
00
m!"' (~a 01\ ~ Im=l
oM 1 0
This completes the proof.
II
for all a
Admf(~)
·m!"'_1\ oV
3
S9
Holomorphic mappings between locally convex spaces
Since every locally bounded polynomial is continuous, we also have shown the following:
E and
Let
Corollary 2.9
U be an open subset of
F be arbitrary rocarry convex spaces.
E and suppose
U and every positive integer
f
E
m,
H(U;F). A
I'
dmf(O
(}l(mE;F)
E
Let ~
Then for every and dmf(~) E
in
l. s (mE;F). Let
Corollary 2.10
be open in
E and
f
E and E
F be arbitrary rocarly convex spaces,
Then if
HG(U;F).
U
is localry bounded it lies in
f
H(U;F). Proof
Apply lemmata 2.7 and 2.8.
We let
denote the vector space of arr Gholomorphic locally
HLS(U;F)
bounded mappings defined on the open subset U of the rocarry convex space E with values in the locally conVex space HLS(U;F) CH(U;F)
for any
U,E
and
F.
F.
We have just seen that
We now look at the reverse
inclusion. Lemma 2.11
If
U is an open subset of a rocally convex space
for every locally convex space
H(U;F) = HLS(U;F)
F
then
E and
E is a normed
linear space. Proof
It suffices to take
identity mapping (from
E
F=E
and to note that the restriction of the
into itself) to
locally bounded if and only if
E
U is always continuous but is
is a normed linear space.
There are, however, several nontrivial examples of pairs of spaces and
F
ings from
E
into
F
coincide.
One can obtain some information about this
problem by extending Hartogs' theorem to locally convex spaces. f; U7G
where
U is an open subset of
E x F,
E, F
convex spaces, is separately holomorphic if for each y
~
E
for whichthe holomorphic and the locally bounded holomorphic mapp
f(x,y)
is holomorphic and for each
is holomorphic.
y
in
F
and x
A function
G being locally in
E the function
the function
x
~
g(x,y)
Hartogs' theorem in several variables implies that separ
ately holomorphic functions are Gholomorphic. Proposition 2.12
Let
E
and
F
be locally conVex spaces and suppose
60
Chapter 2
every separately holomorphic function defined on an open subset of is holomorphic. Proof
Let
f(x,~)
esi~
Then
HLB(U;F)
f s H(U;F). f
~(f(x)).
We define Since
locally bounded function.
f; UxFS ) II:
F,
by the formula
is a locally compact space
II:
Hence, if
~
and a neighbourhood of zero in
subset of
II fllv
such that
x BO
s U,
F ' S
= M <
of
F.
in
~
F'
f
is a
we can find a neighbourhood of BO,
=.
where
Hence
~
for every
Ex FS U of E.
for every open subset
is obviously separately holomorphic and by our hypoth
it is holomorphic.
~,v~,
= H(U;F)
and by Mackey's theorem
f(V~)
B is a bounded
sup IHf(x)) I < = xsV ~ is a bounded subset
This complets the proof.
Example 2.13
If
strong dual of open subset
F
U of
E is a Frechet space and is a Frechet space), then E.
F
is a
HLB(U;F)
DF
space (the
= H(U;F)
for any
This follows from proposition 2.12 since it is
known that separately holomorphic functions defined on open subsets of the product of Fr~chet spaces are holomorphic. Example 2.14
If
E is a ;:, J 8
space and
F
is an :; J
space then
HLB(U;F) = H(U;F) for any open subset U of E. This also follows from proposition 2.12 and the fact that separately holomorphic functions defined on open subsets of a product of fj"1 ~ spaces are holomorphic. We do not know if the same result holds for Jj J
1rz
spaces although we
do know that separately continuous polynomials defined on a product of JJ J ttl spaces are continuous. We now look at functions which are holomorphic analogues of the hypocontinuous and the Mcontinuous polynomials defined in section 1.2 Definition 2.15
A function
localZyconvex space
E
to be hypoanalytic if subsets of
E.
We Let
tic mappings from
f
defined on an open subset
with values in a locally convex space
U of a F
is said
it is GhoLomorphic and continuous on the compact HHy(U;F)
U into
denote the vector space of all hypoanaly
F.
Some authors give a slightly more general definition of hypoanalytic functions  they consider functions which are Gholomorphic and bounded on the compact subsets of
U.
The following example shows that this can lead
to a strictly larger class of functions.
61
Holomorphic mappings between locally convex spaces
Example 2.16 (en)n
Let
E be an infinite dimensional Hilbert space and let
be a sequence of mutually orthogonal unit vectors in
I : (E,a(E,E'))
II Ib
(E,
>
E.
Then
maps compact sets onto bounded sets but is not
hypoanalytic since the sequence
(en)n
is weakly but not strongly
conver
convergent. If
E
is a kspace (in particular, if
a ,'D1fiL space) then
HHy(U;F) "H(U;F)
any locally convex space
locally convex space
is metrizable or if
E
U of
is
E and
F.
A function
Definition 2.17
E
for any open subset
(E,T)
defined on a TM
f
open subset
U of a
is said to be Mackey or Silva holomorphic if We let
~(U;F)
denote the vector
space of all Mackey holomorphic mappings from
U into
F.
it is Gholomorphic and Mcontinuous.
The following result gives an alternative definition of Mackey holomorphic functions.
We omit the proof.
Proposition 2.18
space
Let
E and let
U be a
open subset of the locally convex
TM
F be a locally convex space.
If
f E H (U;F) G
following are equivalent: a)
f E ~(U;F),
b)
for each exists
~ E
E>O
U and each bounded subset B of E there such that f(~+EB) is a bounded subset of F,
c)
for each
~
d)
for each
~EU
E
U
and each and
m
in
N,
g E H(V; E),
H(U;F)
(E,T) for any
d f(q
E
iO m " \f M( E;F).
where ([.
TM = T and hence ~(U;F) E and any locally convex space F.
is a superinductive space then TM
open subset
~(U;F) = H(U;F) .B J,g space and F is for.JJ Jl'rl spaces.
U of
In particular,
if
space or a
arbitrary.
holds
I'm
V is a neighbourhood the function fog is holomorphic
of zero in «: and g(O) = ~> on some neighbourhood of zero in If
then the
U is an open subset of a Frechet We do not know if this result
There are several other types of holomorphic functions to be found in the literature.
We shall introduce them if the need arises.
Our main
62
Chapter 2
interest lies in the study of holomorphic functions and all other function spaces are introduced solely to help our study in this direction.
The diff
erent kinds of holomorphic functions we have defined satisfy the following inclusions.
E and
Let open subset of
E.
F be locally convex spaces and
U an
The following inclusions hold
An important question which will arise in this book and which is still undergoing active research is the following:
for what
some (or all) of the above inclusions proper? this question for polynomial mappings.
U,E
and
Fare
We have already looked at
For the moment, we consider only a
few simple examples. Example 2.19 ~(U;F)
If
E has property
= HHyCU;F)
(s)
for any open subset
and U of
F
is arbitrary, then
E.
The proof given for
polynomials in chapter 1 can be extended. Example 2.20 of
A Gholomorphic function
E with values in
F
f
defined on an open subset
U
is hypoanalytic if either of the following condit
ions hold: a)
f
is bounded on compact sets and
I'm
n
m
d fC~) £ ~HY( E;F) for every and every positive integer m.
b)
f
is bounded on compact sets,
separable and criterion (i.e.
Since
in
U
(E,oCE,E'))
T
and
define the same conver
E).
E is locally convex and hypoanalyticity is a local prop
erty, we may suppose without loss of generality that anced and that a)
Let
is
E satisfies the Mackey convergence
gent sequences in Proof
~
F
KC U
Hence there exists
U is convex and bal
is a normed linear space. be compact. \>1
such that
We may suppose that AK
c: U.
K
is balanced.
The Cauchy inequalities imply
63
Holomorphic mappings between locally convex spaces ~
1/
sup( lid f(O) II ) n ~ 1 < l. A n .l K ~ d f(O) £ ~Hy(nE;F) and hence f is the uniform limit on K By hypothesis nl of a sequence of continuous polynomials. Thus f is continuous on K and lim m+oo
that
f
£
HHy(U;F). b)
Since
By (a) we may suppose that E
be a null sequence in E. such that I A I > +00 and n pact sets
>
00
are metrizable and
(xn)~=l (An)~=l '
Let
There exists a sequence of scalars, Ax n n
F.
n>oo .
as
0
Since
f
is bounded on com
UA m f(x ) n n n
Uf(A x ) n n n
is a bounded subset of
E
is sequentially continuous.
f
>
is an mhomogeneous polynomial.
is weakly separable, the compact subsets of
hence it suffices to show that
n
f
Now
IAnl
00
>
and this implies
f(x n )
>
0
as
and completes the proof. We now look at holomorphic versions of the Factorization Lemma proved
for polynomials in the first chapter.
The situation is much more complicat
ed due to the fact the polynomials are always defined on the entire space and continuity at a single point implies continuity at all points.
These
properties are not necessarily true of arbitrary Gholomorphic functions. Here the topological and geometric properties of the set continuity properties of the function
f
U and the global
have to be taken into considerat
ion. Theorem 2.21
linear space.
Let If
U is a connected open subset of
then there exists an
aCyl = 0
E be a locally convex space and let a
such that for any
cs(E)
£
F be a normed
E and xcU.
YCE
f
£
H(U;F)
for which
and
we have f(x+y) Proof
We first suppose that
is then satisfied if and
f(x).
x
and
U is convex and balanced.
x+y
£
U.
Since
F
Condition (*)
is a normed linear space
there exists an a in cs(E) such that Ba {xcE;a(x)< l} CU M < 00. By the Cauchy inequalities it follows that Ilfll B a
64
Chapter 2
"'m
f'M _ li d f(O) II m! Bet ials we have
for all
.. 1 emma f or po 1 ynomm an db y t h e f actor1zat10n
"dmf(O) m! for every
x
in
x,x+y £ U and
E and all a(y) = 0
=
f(x+y)
(x+y)
m! y
in
) _
,=
E for which
cimf(O) (
cimf(O)
m!
U is convex and balanced.
By the above there exists an
a
f(E;+x) = f(E;+x+y).
a(y) = 0
E;+x+y £ U,
For arbitrary
V a convex balanced open set such that
x+y £ V and x£V,
f(x) .
m!
This completes the proof when E; E U and
Hence if
then
Lm=o    x+y  Lm=O    (x)
we choose
O.
et(y)
then
et(y) = 0
and
in
cs(E)
such that if
U
E;+V CU.
X,y £ E,
Moreover, if
XEV,
x,y £ E,
is satisfied, then we may consider the
(*)
function of one complex variable A
+
f(E;+x+Ay)  f(E;+x)
This function is constant, by the above, on some neighbourhood of zero, and hence it is constant on the connected interval Let
f(E;+x).
U = {x£Ulif y£E, a(y) = 0 o We have just shown that
f(x) = f(x+y)}. U.
If
since
xe E Uo + X E U as e + =, yEE is a topological vector space,
E
ly large and all
A £ [0,1].
Hence
and
[0,11. (*)
Hence
f(E;+x+y)
is satisfied then
Uo is a nonempty open subset of and {x+AyIOl"Al"I} CU, then) Xe+AY £ U for all
e
sufficient
f(x+y) = lim f(xe+ Y) = lim f(xe) = f(x) 8+=
Thus
Uo is a nonempty open and closed subset of U. ected, this implies U=U and completes the proof. o
8+=
Since
U is conn
Our next step motivated by the polynomial case would be to define on
na(U)'
f
There are, however, several difficulties which cannot be sur
mounted without certain modifications. be well defined.
Without condition
(*)
rv
f
may not
It is possible to surmount this problem, in the general
situation, by considering domains spread over
E and using a pullback oper
ator or by restricting oneself to pseudoconvex open sets. presentation, we confine ourselves to convex open sets. arises from the fact that
na(U)
To simplify our
A second difficulty
is not necessarily an open subset of
E
65
Holomorphic mappings between locally convex spaces
However,
ITa(U)
will always be a
we can ask, assuming
t f open subset of Ea is well defined, whether or not
1
and consequently
f
is a holomor
,."
phic function.
The set of points of continuity of
may not be all of
ITa(U).
f
will be nonempty but
Example 2.22 illustrates this difficulty.
difficulty is overcome by placing extra conditions either on mappings
Example 2.22
We denote by
the space of entire functions of one
H(~)
complex variable endowed with the compact open topology. be defined by
a
subset of
= f(f(O)).
F(f) H(II:).
function through exist an
This
or on the
and we give various sufficient conditions.
IT a
function on
Ea
in H(t)a.
It is easily seen that
We claim that
Let F
F:H(()
+
II:
is an entire
F does not factor as a holomorphic
H(()a for any a E: cs(H(II:)). Otherwise, there would cs (H(I[)) such that B {fE:H(II:);F(f) = O} is a closed Without loss of generality we may assume
a(f)
suplf(z)1
where
R> 2.
Izl~R
For each positive integer
n
let
z+z 2 +... + zn (2R) + ... + (2R)n Then
fn (0)
= 0,
o.
fez) Since If
(*)
2Rz2. F(f)
Then
f(2R)
2
2R).
= 4R2  2R for all nand
fn (2R)
sup If (z) I I zkR n Let
(z
=
+
0
F(f+fn) 2R4R 2 F 0
as
n
+
00.
(f+fn)(f(O)+fn(O))
=
f(2R)+f n (2R)
we have shown that no such
U is convex and balanced (or even pseudoconvex)
of Theorem 2.21 is satisfied for all
x
and
y
a
exists.
then condition
and we obtain the
following factorization result. Proposition 2.23 normed linear space.
Let If
E be a locally convex space and let
F be a
U is a convex balanced open subset of
=
E and such
Chapter 2
66
TIa(X) = x. Theorem 2.2. shows that f is well defined and by construction ,... f = foTI . Since f is a Gholomorphic function, it follows that a
We now give a sufficient condition for the continuity of
If
Proposition 2.24
(i.e.
if
U
H(TIa (U) ; F)
in
a
D and
fa s H(lla(U);F) U of
E and
F.
We first note that in the proof of theorem 2.21 we may choose
Proof to lie in
D.
Hence
lla(U)
is an open subset of
proposition 2.23, fa s HG(TIa(U);F) locally bounded since there exists a
F
B in
II £\1 B
cu and since
a
ascs(E)
for any convex balanced open subset
faolla)
II
D then
in
then there exists an
f s H(U;F) f
a
U H(ll (U) ;F) asD a
=
any normed linear space
u
contains
cs (E)
E is a locally convex space and
is an open mapping for every
such that
f.
D of seminorms which define the topology of E and
a directed set
H(U;F)
,...,
<
such that
and Hence
00
BS
fa
Now
f
is ~
Hence for each
in
= {xsE;B(t;x)
c
)s,p
11r:ll ll (B ) a S,t;,p
is open, it follows that
to H(lI a (U);F). obvious.
such that
p>O
and there exists, by
f = faolla'
is a normed linear space. D
B,t;,p
lla
Ea
a
Ilf[I B < S,t;,p
00
and,
is locally bounded and so belongs
This completes the proof since the opposite inclusion is
The above proposition covers the case where each
E a
is a Banach
space and yields the following examples. Example 2.25 Let
Ea
Let n
n
= II j =1 Ej
00
E
11
for all
n=l n.
E where each En is a Banach space. n If f s H(E;F) and F is a normed
linear space, then there exists a positive integer such that Ea
nand
f n s H(E
where lln is the canonical projection of f = f 011 n n As a particular example, we see that
n H(a;N)
VN
H(a;n).
;F)
an E onto
67
Holomorphic mappings between locally convex spaces
Let
Example 2.26 and let
= ~
E
Ea ~ t(K),
(X).
X be a completely regular Hausdorff topological space Then we can choose our directed set
K compact in
for each
X,
a
in
D such that
Hence each
D.
Ea
is
a
Banach space and we obtain a factorization result for normed linear space valued holomorphic functions defined on convex balanced open subsets of
.& (X)
•
A further sufficient condition, this time on the spaces in the following proposition.
Ea'
is given
This condition also arises in various other
parts of infinite dimensional holomorphy, for instance analytic continuation and we show (theorem 2.28) that it is satisfied by any Banach space. fact, we prove a more general result which we shall use later. gives an alternative proof for examples 2.25 and 2.26.
In
This, then,
In more general
factorization theorems it is applied to prove results unobtainable by using proposition 2.24. Proposition 2.27
Let
normed linear space.
E be a locally convex space and let
If there exists a directed family
which define the topology of condition:
E
and each
in
a
F be a
D of seminorms
satisfies the following
0
if
f £ H(Ua;F) where Ua is an open subset of Ea then the set of points of continuity of f is open and
cLosed. Then
u
H(U;F)
H(IIa(U);F)
a£cs(E)
for any convex balanced open subset Proof
E.
This follows immediately from theorem 2.21.
Theorem 2.28
let
U of
U be a connected open subset of the Banach space E, F be a normed linear space and let f £ HG(U;F). If dmf(1;) £ fr> (mE;F) Let
for some
I:;£U
and every positive integer
Proof
Without loss of generality, we may assume that
balanced set and that
I:;
= O.
Let
m then
f £ H(U;F). U is a convex
Chapter 2
68 ~
Vn
{XEU;
Id :~O)
(x)
I;; n
for all
m}
,...
dmf(O)
Since each
is continuous,
Vn
is a closed subset of
U and
00
U V = U since f is Gholomorphic. By the Baire category theorem n=l n has nonempty interior. If TjEU and V is a convex some Vn ' say V n0 ' then lemma l.13 implies balanced neighbourhood of zero such that n+VCVn 0
that
VCV
no
II fill
Hence
rl "
dmf(O)
sup 2XEV
;;
"IV
n0 m 2
00
L m=O
;;
Thus
f
m!
m=O
(x)
I
2n 0
is locally bounded and continuous at the origin.
By using the
Taylor series expansion of Gholomorphic functions, we see that 1\
A
00
n d f(6) (x)
2.
for any
8
in
for any
n
and any
[d
I'm
n
d f(O) ) (6)] (x) m!
m=n
U and any 8
x
in
U.
that
in
E.
By lemma l.19,
ci nf(8)
E
61 (nE;F)
By the first part of our proof, i t follows U we
f is continuous at 8. Since 8 was arbitrarily chosen in have shown that f EO H(U;F). This completes the proof. Since Frechet spaces and
d2J1J
spaces are superinductive limits of
Banach spaces, one can easily prove a result similar to theorem 2.28 for such spaces. So far we have been describing factorization results which use continuous seminorms on the domain space. sol ving the Levi problem on sort of factorization. A topological space
Certain situations (for example, in
dJ '1nt. spaces
with a basis) require a different
We give some results in this direction. X is a Lindelof space if every open cover of
contains a countable subcover.
X is said to be hereditary Lindelof if
X
69
Ho!omorphic mappings between locally convex spaces
every open subset of
Separable Frechet spaces and ~ J'h1
X is Lindelof.
spaces are examples of hereditary Lindelof locally convex spaces.
Let
ProEosition 2.29
U a convex
F a normed linear space and f r:: H(U;F)
then there exist
depends on
f
N
f
Proof
UJ,
and
H(I1 f (U) ;F)
£
such that
For every
~s
~ocaUy
a metrizable
U there exists an
II f 111;+ B .
ba~anced
convex space,
open subset of E.
If Ef (which E onto ~ and and f = fo I1 f .
convex space
a continuous surjection I1 f from I1 f (U) is an open subset of Ef
in
I;
Linde~of ~oca~~y
be a hereditary
E
al; r:: cs(E)
such that
(1) < where B (1) = hr::E;a c (x) < l}. al; " al; an open cover of U it contains a countable sub00
00
The seminorms
(al; )n=l
generate a semimetrizable
n
locally convex topology on
E.
Let
Ef denote the associated metrizable space and let I1 f denote the quotient mapping from E onto Ef . By construction I1 f (U) is an open subset of Ef. We now define on I1 f (U)
r
in the usual manner and since it is locally bounded, it lies in
H(I1 f (U);F).
This completes the proof. Corollary 2.30 E
and
If
is a convex ba~anced open subset of a J:; J 'ht space
U
is a Banach space, then
F
u
H(U;F)
H(I1a (U) ; F) .
ar::cs (E) A b1rQ space is a hereditary Lindelof space and also a DFspace.
Proof
The result now follows easily by using the construction of proposition 2.29 and the following property of
DF
spaces:
if
(an)~=l
is any sequence of
continuous seminorms on a DFspace, then there exists a continuous seminorm an
~
a c na
on
E and a sequence of positive real numbers
for all
(cn):=l
such that
n.
Corollary 2.30 may be strengthened in the case of entire functions (see exercise 2.105). Our final example first arose in finding a counterexample to the Levi problem.
The proof is quite different from those just given and variations
of the technique used will appear in chapter 5. ExamEle 2.31
Let
r
denote an uncountable discrete set.
If
Chapter 2
70
x = (xa)ad E co(r)
sex) = {ad ;xa f O }.
Let
co(f 1)
fl of f. Now suppose f E H(2B;') where B is the open unit ball of (co(r), II II)· {al' ... ,a n } is any finite subset of f then, by using a monomial
{XECO(r);S(X) Cf l } and II fll B=M
we let
J
for any subset
expansion in several variables, we see that fez)
for all
f(
f
dz.
in
2B
co(J).
0
ith coordinate equal to 1 and all other coordinates 2 > sup{lf(z)1 ; z = L z.e·,lz·l= 1 an i} i=l l l l
e i has its zero, then M2
If
where
z
L z.e.)f( f z.e.) i=l l l i~l l l n
'I'n
z
n
is normalised Haar measure on {z.e.,1 z. 1=1} for i=l, ... ,n. iSm l l l zm = e , m=l, ... ,n, it follows that
l
By using the change of variable
Since
JC:f
was arbitrary, it follows that
2
M>L (r)iw(k)i hN where
N(r) = {¢
Hence
{k;w(k)fO} Let
:
f .... N,
fl = {aE:f1
able subset of
f.
¢
(a)=O
2
for all except a finite number of
a
in
n.
is countable.
3
k EN(r),
w(k)fO
It is easily seen that
and
k(a)fO}.
fl
f(x+Ae) = f(x) a
U
is a countfor all
(x,A), x E 2B and x+Ae a E 2B, if a £ f,f l · Since JeT,J finiteco(J) is a dense subspace of co(r) (in the norm topology) and f is continuous, we have shown that (Xa)aEf
E B.
f( (x) ) = f( (x ) ) for all a aEf a aEf l By using the principle of analytic contlnuation (in several
71
Holomorphic mappings between locally convex spaces
complex variables) one can easily extend the above proof to show the following: if
U is a convex balanced open subset of
co(r)
onto
r"
f
=
Co (r 1) .
foIl
rl
where
ITr
f
E
H(U;OC)
r l of rand f E H(U" co(r l );([) is the canonical surjection of co(r)
then there exists a countable subset such that
and rJ
1
Many of the above factorization theorems can be extended to pseudoconvex domains (and this essentially means to all open sets) by virtue of the following result: if and
U is a pseudoconvex open subset of the locally convex space
U contains an aball, a
finitely open subset of
E a
E
cs(E),
and
U
=
then II (U) 1 11 (11 (U)). a a
E
is a pseudoconvex
This result is used in &udying pseudoconvex domains, holomorphically convex domains and domains of holomorphy in locally convex spaces. Factorization results for Mackey holomorphic functions are required in Chapter 6.
The concepts and methods needed to prove these results will be
given later. §2.3
LOCALLY CONVEX TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS Two topologies are usually considered on the space of Mackey holomor
phic functions;
the topology of uniform convergence on the finite dimen
sional compact subsets of the domain and the topology of uniform convergence on the strictly compact subsets of the domain.
Since we shall not use
any results derived by using these topologies, we confine our interest in these topologies to some exercises at the end of the chapter.
On the
space of hypoanalytic functions, the natural topology is the compact open topology. Definition 2.32 E
and let
F
Let
U be an open subset of the locally convex space
be a locally convex space.
The compact open topology (or
the topology of uniform convergence on the compact subsets of UJ HHy(U;F)
on
is the locally COnvex topology generated by the seminorms
72
Chapter 2
II fll s,K
sup S(f(x)) XE:K
K ranges over the compact subsets of U and S ranges over the F. We denote this topology by TO.
where
continuous seminorms on Naturally
TO
induces a locally convex topology on
again call the compact open topology and denote by
H(U;F)
which we
This is the most
TO.
natural topology to consider on spaces of holomorphic functions.
We find,
however, it does not always possess very useful properties and for this reason we introduce the
topology.
TO
This topology has good topological
properties but can be difficult to describe in a concrete fashion and its relationship with the
topology may not always be clear.
TO
duce a further topology,
T
w
We also intro
whose definition was motivated by certain
,
properties of analytic functionals in several complex variables. topology is intermediate between the
TO
and the
fully it will share the good properties of both
TO TO
This
topologies. Hopeand TO but its main and
role appears to be as a tool in proving results about
(see
TO
for instance chapter 5). Definition 2.33
and let
Let
U be an open subset of a locally convex space
F be a normed linear space.
there exists p(f) ~ C(V)
The
C (V»
for all
let
F
Let
H(U;F)
is said
f
E
H(U;F).
is the locally convex topology generated by
all seminorms ported by compact subsets of Definition 2.34
on
a such that
IIfllv
'w topology on H(U;F)
p
K of U if, for every open set
to be ported by the compact subset V, K eVe U,
A seminorm
E
u.
U be an open subset of a locally convex space and
be a locally convex space.
We define
'won
H(U;F)
by
lim SE:cs(F)
Definition 2.35
and let
Let
U be an open subset of a locally convex space
F be a normed linear space.
A seminorm
p
on
H(U;F)
E
is said
73
Holomorphic mappings between locally convex spaces
to be
TO
continuous if for each increasing countable open cover of U,
(Vn)~=l'
there exists a positive integer
no
II fllv
f
P (f)
c
~
for every
topology on
the
To
continuous seminorms. Let
H(U;F).
is the locally convex topology generated by
H(U;F)
TO
Definition 2.36
in
such that
n0
The
and let
and c ,0
U be an open subset of a locally convex space
F be a locally convex space.
We define
To
on
H(U;F)
E
by
lim SECS (F)
The general relationship between the three topologies just defined is given in the following lemma. Let
Lemma 2.37
E and F be locally convex spaces and let
open subset of E. Proof
H(U;F)
we have
We may suppose, without loss of generality, that
linear space. follows that on
On
H(U;F)
Since Tw
~
II fll K
TO
~
II fllv
Now suppose
for every p
is a
exists
c>O
shows that
F
V containing TW
which is ported by the compact subset
denote an increasing countable open cover of can choose
U be an
U.
is a normed K it
continuous seminorm K of Since
U.
Let
(Vn)~=l
K is compact we
no such that Vn 0 is a neighbourhood of K. Hence there such that p (f) ~ c II fll V for every f in H(U;F) . This no p is TO continuous and completes the proof.
Our next result shows that
TO
has good topological properties when
the range space is a Banach space (a slightly less general result holds when the range space is a normed linear space). alternative description of the Proposition 2.38
Let
TO
This result also gives an
topology.
U be an open subset of the locally convex space
E and let F be a Banach space. Then (H(U;F),T O) is an inductive limit of Frechet spaces and hence it is barrelled, bornological and uZtrabornological.
74
Chapter 2
Proof let
For each increasing countable open cover of
Ht:>(U;F) = {fEH(U;F); Ilfllv
all
< ""
n
, We endow
nL
Pn (f) = II fl~ .
topology generated by the seminorms metrizable locally convex space.
U\...9 = (V)""
n n=l' HiJ(U;F) with the
H~(U;F)
is then a
It is in fact a f~echet space since
F
is a Banach space and locally bounded Gholomorphic functions are holomorphic.
H(U;F) = LJ~H~(U;F)
We claim
countable open covers of Wn
(Wn)~=l
is open and UY
Since
U.
f E HU9 (U;F) (H(U;F)"i) =
follows that
ranges over all increasing
f E H(U;F) we let
{XEU; Ilf(x)11 < nl Wn countable open cover of U. is an increasing
we have proved our claim.
inductive limit topology on i.e.
If
where ~
H(U;F)
~ H~(U;F).
(H(U;F)"i)
We now let
,.
~
defined by all the spaces H~(U;F)
Since
denote the Hc;(U;F) ,
is a Frechet space it
is an ultrabornological space and hence it is
barrelled and bornological. We complete the proof by showing that mapping from
H~(U;F)
able open cover Let not of
II fnllv
denote a
p
H(U;F}
Since the identity coun~
is continuous for any increasing
U we have 'i continuous seminorm on H(U;F}. Suppose p is Then there exists an increasing countable open cover
continuous.
'0
U,
into
of
~
'i = '0.
and a sequence (fn}~=l (Vn}~=l = \J I and p(f ) > n for all n. n
in
H(U;F)
such that
~
n
Let
Wn = {XEU; Ilfm(x)II ::: n
W~ denotes the interior of Wn .
for all
m}.
Let 0
Ii fm II V ~
Since
I
= (W~)~=l where for all
follows that ~ is an increasing countable opfn cover of
U.
me:n,
it
Hence
pIHU9 (U;F) is continuous. By construction s~p Ilfml~!ii n for every n. Hence (fn}~=l is a bounded subset of H~(U;F). This gontradicts the fact that
p(fn )
>
n
for all
n.
Hence
'i = '8
and we have completed
the proof. and
Since 'o,b with H(U;F)
and '0
,
w,b and 'w
,
w
are not in general bornological topologies, we let
denote the bornological topologies on respectively.
H(U;F)
associated
(Note that the topology induced on
by the bornological topology associated with the compact open
topology of
HHy(U;F)
need not be
ogies that can be placed on
H(U;F)
,
0,
b).
There are also further topol
such as the topology of uniform
convergence of the function and its first
n
derivates on the compact
75
Holomorphic mappings between locally convex spaces
subsets of
U, n = 1,2, ... ,00
but since we shall not use these topologies
we will not go into any further details. A great portion of this book is concerned with finding conditions on
U,E and F which imply either ' 0 = 'w' ' 0 = '8 or 'w '8 (together with the implications of these conditions). The remainder of this section is devoted to a number of basic facts, concerning these topologies, which we shall frequently use and to a few examples and counterexamples which will prove useful in later chapters. We first note that the compact open topology is a sheaf topology, i.e.
'8
,w
We do not know if this is true for the
it is locally defined.
and the
topologies and any results in this direction would certainly help the The local character of the compact open
general development of the theory.
topology is contained in the following lemma. Lemma 2.39 (Ui)iEI
Let
U be an open subset of a locally convex space
an open cover of U and
F a locally convex space.
E,
The mapping
from (H(U;F)"o) into ITiEI(H(Ui;F)"o) which maps f to (flu.)iEI l. (where flu. is the restriction of f to Ui) is an isomorphism of H(U;F) Proof
l.
onto a subspace of ITiEIH(Ui;F) It suffices to note that
K is a compact subset of
only if there exists a finite subset of a compact subset of
U,Q,
for each
j,
j
I,
,Q,l, ... ,,Q,n
such that
and
K =
U if and n
(Kj)j=l'
Kj
U ~ =1 Kj .
It is obvious that a similar result holds for hypoanalytic functions. We now show that (H(U;F)"o)
(fr>(mE;F)"o)
for any open subset
complete local,ly convex space
F,
is a closed complemented subspace of
U of the locally convex space and any positive integer
m.
E,
any
Hence if
(H(U;F)"o)
has any property inherited either by subspaces or by quotient
spaces then
(cp(mE;F)"o)
Proposition 2.40 and
F
If U is an open subset of a locally convex space
is a complete locally convex space then
complemented subspace of Proof
must also have the same property.
Since
(H(U;F)"o)
UP (mE;F), '0)
for any positive integer
(H(U;F) "0) = (H(Ui;;,F) "0)
for any
I;
E
is a closed m.
E E we may suppose
76
oE
Chapter 2
U.
As uniform convergence on the compact subsets of
E
is equivalent to
uniform convergence on the compact subsets of some neighbourhood of zero for
tS> (mE;F)
elements of
i t follows that
the compact open topology.
dm
induces on (f>(mE;F)
Now consider the mapping
H(U;F)
m!
(H(U;F)"o)
H(U;F) " dmf(O)
f
m! Am This is a linear mapping and since ; (P) (0) P for every P in m. l.P(mE;F) it is a projection from H(U; F) onto lP (mE; F): To complete the proof we must show that it is a continuous projection. convex balanced open subset of balanced subset of
E
such that
V CU.
Let If
V denote a
K is a compact
V then, by the Cauchy inequalities
"
II dmf~O) I s,K m. for every
S
in
cs(F)
and hence the projection is continuous.
This
completes the proof. For arbitrary
U we do not have any useful representation of the
topological complement of
~ (~;F)
in
H(U;F)
but we shall see, in the
next chapter, that the Taylor series representation of holomorphic functions gives us a means of identifying a useful topological complement when balanced. '8
We now prove the analogue of proposition 2.40 for the
topologies.
U is
'wand
Our proof is for Banach space valued mappings but the same
result for an arbitrary complete locally convex range space can be proved in a similar fashion.
If U is an open subset of a locally convex space E and F is a Banach space then (iP (mE;F)" ) is a closed complemented w subspace of (H(U;F)" ) and of (H (U; F) ,T 8) . In particular and '8 w w induce the same topology on 6> (mE;F). ProEosition 2.41
,
Proof
We first show that
denote a
'8
,
and w continuous seminorm on
balanced neighbourhood of zero in increasing countable open cover of integer
Nand
C>0
such that
E. U
'8 coincide on H(U;F) and let
19 (mE;F). V
Let
(U" nV) ~=l is an and hence there exist a positive
The sequence
p
denote a convex
77
Holomorphic mappings between locally convex spaces
':'m
for all
~
P ( d f (0)) m!
p(f)
f
II dmfm!(0) II
U", NV
H(U;F).
€
p
Hence
c
pi
is a
T
continuous seminorm on
w
= mP\iPCmE;F) it follows tha t
(p(mE;F) topology on
T6
and since
H(U;F)
and T w indue e the same
The above also shows that the mapping given in proposition 2.40 is a continuous projection for both
Tw
and
T
6
•
We now look at the locally bounded or equicontinuous subsets of H(U;F). Definition 2.42 E
and let
Let
U be an open subset of the locally convex space
F be a locally convex space.
locally bounded if for every
s'Vs'
s
in
A subset J
of
H(U;F)
is
U there exists a neighbourhood of
such that
is a bounded subset of F. Lemma 2.43 A locally bounded subset of H(U;F), U an open subset of E where E and F are locally conVex spaces, is a bounded subset of (H(U ;F) ,T 6 ). Proof
We may assume, without loss of generality, that
linear space. TO
Let
:J
be a locally bounded subset of
continuous seminorm. Wn
and let
Vn
For each positive integer
{X€U; Ilfex)!! Interior (Wn ) .
~
n
Since
for every
J
f
F
is a normed
H(U;F), n
and
p
a
let
in j}
is locally bounded
(Vnh
is an
Chapter 2
78
increasing countable open cover of
U.
Hence there exists
C>O
and
N,
a positive integer such that p(f) ;; C 1\ fl\ V
for every
f
in
H(U;F).
N
Hence
sup p(f) ;; C.N fE J
and this completes the proof.
If
Corollary 2.44
U is an open subset of a LocaLLy convex space
is a locaLLy convex space and every bounded subset of locaLly bounded then
T
0'
T
T~
and
Ul
(H(U;F)
E, F
is
,TO)
have the same bounded subsets in
v
H(U;F).
In particular Notation
is the bornoLogicaL topology associated with
TO
etc. in place of
Corollary 2.45
E and
Let
bounded subsets of
(H(U)
of E then
=
Proof
H(U;F)
linear space.
f(K)
,TO)
H(U;F ) a
~y(U;[),
H(U;[),
= =
•
F
etc.
be LocaLly convex spaces.
Let
where
=
Fa
(F,a(F,F')).
B be the unit ball of
(epof) epEB ep(f(K))
lies in
H(U).
If
Thus
J
F
Fe
F
is a normed
f E H(U;F ). The a K is a compact subset of U then
is a bounded subset of
is a weakly bounded subset of
strongly bounded.
If the
are LocalLy bounded for every open subset U
We may assume without loss of generality that
epcf(K)
o
If the range space is the field of complex numbers we write
H(U), HHY(U)
set j.
T
[
and let
for every
ep
in
F'.
Hence
and by Mackey's theorem it is
is a bounded subset of
(H(U),T
O
)
and
50
by
our hypothesis, it is locally bounded. Hence for each E, E U there exists an open set containing E, such that supllepofll v <"'. epEB E, Consequently is a weakly bounded subset of F and once more f(V~)
by Mackey's theorem it follows that f
is a locally bounded function.
Hence
is a bounded subset of f
F, i.e.
is a holomorphic function and
this completes the proof since the composition of holomorphic functions is holomorphic and so we always have Example 2.46
Let
space and let subset, E,EU
2r,
F of
H(U;F)
C H(U;F a)'
U be an open subset of a metrizable locally convex
be a normed linear space. H(U;F)
is locally bounded.
such that for every open set
V,
E,
We claim that any
TO
bounded
If not, then there exists
E veU,
wehave
supllfll v
fd
=
00
79
Holomorphic mappings between locally convex spaces
Hence we can choose
~JI E
U,
~n ~
~
as
n
~
(f) C n n
and
00,
II fn (~n) II
J such
that
U
is a compact subset of > n for all n. Since {~n}n u {O this is impossible and we have proved our claim. The remaining examples given in this section deal with holomorphic functions on J):f
m
spaces, Banach spaces,
([N x G: (N)
and on locally
convex spaces which do not admit a continuous norm.
These examples are
elementary insofar as the proofs are rather direct.
However, they are of
interest since they show the divergence between linear and holomorphic functional analysis and also because many of the examples and methods encountered here have explicitly and implicitly motivated the development of the theory as outlined in this book.
These examples also provide a good
intuitive guide to the type of behaviour we may look for in delicate situations. U be an open subset of a Jj j h1.
Let
Example 2.47
F be a normed linear space. H(U;F)
We show that the
E and let
are locally bounded (we have already proved this result for homo
geneous polynomials in chapter 1). ity, that TO
space
bounded subsets of
TO
U is a convex balanced open subset of
bounded subset
(Bn)n
We may assume, without loss of general
3F
of
H(U;F)
E and we show that the
is locally bounded at the origin.
be a fundamental system of convex balanced compact subsets of
Let E.
As we have previously noted, it suffices to find a sequence of positive real numbers,
Since
(An)n'
such that
is compact, we can choose M < 00. Now suppose sup II filA B fE:3kl 1 L
Ii=l \Bi
Hence
such that
Bl
have been chosen so that
is. a compact subset of
U
and
supllfll L ~ M
If
0>0
choose and
we let
°1 > 0
8 2 >1,
L(o) = L such that
such that
~ =M' Ln=l 2n .
+ ,k
fE .J"
+ oB + . Let E >0 be arbitrary. We first k l L(ol) C U and next choose 81 and 8 2 , 81 > 1
8 82L(01) 1
is also a compact subset of
U.
p(f)
for all
f
in
inequalities.
H(U;F) Hence
where
c(8 l )
is derived by using the Cauchy
Hence
80
Chapter 2
is a
p(f)
continuous seminorm on
H(U;F)
we can find a positive integer sup \'L 00 _ f£1 nN+l
Let
J
and since N
is
T
o
TO bounded
and
13 >1
2
such that
  II II ~nf(O) n! L('\)
~ E/ 4 .
be the symmetric nlinear form associated with dnf(O)/n!. n!
If
f E H(U;F)
and 0
>
then
0
for any nonnegative integer Since
sup fd
II
dnf(O) n!
n.
II
<
L(o)
~
for each
nand
0 we can choose
so that
N
sup 02 fd· Hence
In=l sup xEL,YEB k + 1 sup fE J
II fll
L (0 2 )
II ~
~
n Ij=l ( ~J ) sup fE :1
dn~(O)
(x)n j (02 y )jl(y)
II
<
E/2.
n. dnf(O)
I:=o
II
n!
IIL(o ) 2
M'+ £/2 + £/4.
Hence, by induction, we can choose a sequence of positive real numbers,
(f'n)~=l'
iT
sup II fll '" ~ M+ 1. Hence f£:1In=l AnBn is a locally bounded family of functions. This implies that TO' TW such that
1':=1 AnBn C U and
and TO define the same bounded subsets of H(U;F). Since 3:J'J"IYt spaces are hereditary Lindelof spaces and contain a fundamental sequence of compact sets it follows that every open subset of a ;;0:1 h1.. space contains
81
Holomorphic mappings between locally convex spaces
a fundamental system of compact sets.
This, in turn, implies that
(H(U;F),T ) is a metrizable, and hence a bornological locally convex space. O Since To is also a bornological topology on H(U;F) we have in fact shown that T0 T = '0 on H(U;F). Finally we remark that open subsets of
w ~Jh1 spaces are
kspaces and so
is a Frechet space if
(H(U;F) ,TO)
F
is
a Banach space.
Banach space (an)~=o E c 0
U be an open subset of an infinite dimensional
Let
ExamEle 2.48 E.
Let
t;
B be the unit ball of
U and let
E
•
f
continuous seminorm on
w
compact open topology on subset
If
(the space of null sequences of complex numbers) then
p(f)
is a
E.
H(U).
H(U)
E
H(U),
which is not continuous for the (H(U),.) F (H(U),T ) o w
Hence
for any open
U of any infinite dimensional Banach space.
ExamEle 2.48
Let
E = eN
x
~(N).
(If> (2E) ,T ) F ((p (2E) " ). Hence o
set of
(N)
(N x II:
We have already seen that
(H(U) ,T ) F (H(U) ,T)
for any open sub
w o w
• Example 1.39 shows, also, that
T F Tw,b o,b
on
H(U).
For our next example we need a concept which frequently arises in infinite dimensional holomorphy  the concept of very strong sequential' convergence  but which does not arise in linear functional analysis,
Since
the dual concept  very weak sequential convergence  will also be needed later, we take the opportunity of giving its definition here.
Further
information on these concepts is outlined in the exercises.
Definition 2.50
A sequence
(xn)n
said to be very strongly convergent if sequence of scalars xn
F0
for each
(An)n'
in a locally convex space AnXn~
0
in
E as
n~
E is for every
The sequence is said to be nontrivial if
n.
A sequence is obviously very strongly convergent if and only if for each
p
in o
csCE) for all
there exists a positive integer, n> nCp).
nCp),
such that
A metrizable locally convex space
a nontrivial very strongly convergent sequence if and only if
E admits
E does not
82
Chapter 2
admit a continuous norm. un
For example, in
[N
the sequence
(0, ... ,1,0 ... ) is a nontrivial very strongly convergent sequence. ~ nth position
Definition 2.51
A sequence
in a locally convex space
(xn)n
to be very weakly convergent if An xn of nonzero scalars
in
+ 0
as
E
E is said
for some sequence
(An)n.
For example, in
[(N)
the sequence
very weakly convergent sequence. is not a Banach space then
E'
un
(0, ... ,1,0, ... )
In fact, if
is not a
~ nth position is any Frechet space which
E
contains a sequence which does not converge
very weakly. Example 2.52
Let
E be a locally convex space which contains a non(xn)~=l .
trivial very strongly convergent sequence For each
f
If
f
then, by the Factorization Lemma, there exists an
H(E)
E
ive integer f(ny)
f
for every
nu
such that ~
n
nu
u(x n ) =
and
p(f)
and
w in
uous seminorm on
H(E)
a barrelled topology on
°
for all
bounded subsets of
defines a
H(E)
it follows that
B be a
H(E), '0
i.e.
p
is a
'o,b
bounded subset of
p
such that
in
B.
n
~
no
and all
f
cs(E) =
o.
f(ny+x n ) =
in
H(E).
The
contin
'0
is
'0
continuous
fn(AnY + xn)
F fn(AnY)
for all
p
is bounded on the
continuous seminorm on We begin by showing that
f(AY+x n ) = f(AY) for all If this were not true, then by
using subsequences if necessary, we can choose that
E
u(w)
nu and since is a
H(E).
no
all
Hence f
and hence a
'0
A
t,
nu.
for every positive integer
there exists a positive integer E
~
n
We now improve this result by showing that Let
U
such that
H(E).
seminorm on
H(E).
E
is finite for every
\f(ny+x n )  f(ny) \
+
x
is a very strongly convergent sequence there exists a posit
for all
function
belong to
we consider the sum
f(x+w) = f(x)
(xn)~=l
Since
H(E)
°
E.
such that
in
yF
Let
n.
An
E
t
and
fn
E
B such
For each positive integer
n
let
83
Holomorphic mappings between locally convex spaces
By the identity theorem for functions of one complex variable we may select a sequence
F O.
gn(A~)
O
f complex numbers,
For each integer
Now
hn s H(() for each n, sequence of complex numbers, for all
n.
n
and each
and since
(wn)~=l ' I f n (A'n y+w n xn ) I > n
Hence
K
all
n
for all
n
it follows that
w s II:
let
hn(O) F hn(l), for all
we can choose a
I hn (w n ) I > n+ If n (A'n y) I n.
Since
(xn)n
A~ y +wnxn
n
as
0
+
p
otherwise, i.e. that
is not a p
'w
continuous seminorm on
and hence
is ported by the compact subset
H(E).
K of
using subsequences if necessary we can choose a sequence in such that and for
m,n n
F 0,
K
in II <pollv
n,
positive integers
II<pollv .
Since
m
follows that
m I
Hence
By
(
for all
n>O
n
and
m.
m. If
V is any neighbourhood ~
c(V) II fllv for every of K such that
11
and
m
ml<po(Y) I ~
E.
E' ,
n=m. Let n+l
then ther~ exists c(V) > 0 such that p(f) H(E). Choose an arbitrary neighbourhood Vm < =
Suppose
i f and only i f
for all
Taking
V
for all
p(
Let
Since
Hence there exists a positive integer no such that f(AY+x n ) = for all all A s I\: and all f in B. This shows that
We now show that
f
is a
is a null sequence.
{O} lJ {A~y+wnXn};=1 is a compact subset of E. As II fnll K > n for this contradicts the fact that B is a ' 0 bounded subset of
H(E).
of
such that
n
such that
very strongly convergent sequence, (wnxn)n I A~I ~ nl2
I An 'I '<: ~ 2'
(A n')=n=l'
p
Vm
II
is not
{fsH(E);p(f)
n
~1}.
,w
th
roots and lettine
n+=
we get
was an arbitrary neighbourhood of This cannot hold for all
K
it
m since
continuous.
V is a convex balanced
(and hence, ') w
84
Chapter 2
bounded subset
of
E.
Since
V is not a
neighbourhood of zero (because that neither spaces.
(H(E)"o)
nor
p
is not
(H(E)"w)
'w 'w
'0)
(and hence not a
continuous) we have shown
are infrabarrelled locally convex
The above can easily be modified to show that the same result
holds for
H(U:F1,
U an arbitrary open subset of
E,
and
F
any locally
convex space.
§2.4
GERMS
OF
HOLOMORPHIC
FUNCTIONS
We now introduce the space of holomorphic germs on a compact subset of a locally convex space.
Apart from its close relationship with spaces of
holomorphic functions defined on open sets, the space
of germs is also an
important tool in developing a satisfactory duality theory.
The problems
that arise in studying the topological vector space structure of the space of germs are of a different kind from those which arise in function space theory and this difference arises primarily from the difference between projective and inductive limits. Let
K be a compact subset of a locally convex space E and let
be a locally convex space.
On
~
H(V;F)
F
we define the equivalence
V:::>K V open
relation which
f
..J
where
and
g
We denote by
f,..... g
if there exists a neighbourhood
W of
K on
are both defined and H(K;F)
the resulting vector space of equivalence classes
and the elements of H(K;F) are called holomorphic germs on K. If f is an F  valued holomorphic function defined on an open subset of E which contains K then we also denote by f the equivalence class in H(K;F) determined by
inductive limit
f.
The natural topology on lim (H(V;F)"w)
H(K;F)
is given by the
(the inductive limit being taken in the
+
v::n
Vopen category of locally convex spaces).
for
If F is a normed linear space we let Hoo(V;F) = {f £ H(V;F) ;11 fllv < oo} V open in E and on this space we define a topology by means
of the norm II II V' HOO(V;F) is a normed linear space which is complete if F is a Banach space. Using the same equivalence relationship we easily see that
85
Holomorphic mappings between locally convex spaces
u
H(K;F) .
V'::'K
V open
If
Lemma 2.53
K is a compact subset of a locally convex space
E and
is a normed linear space then
F
lim
lim
H(K;F)
+
If
Proof
V~K
V:l K
V open
V open
E which contains
V is an open subset of
natural injection from
Hoo(V;F)
the identity mapping from
lim

+
V:JK V open
V:::>K V open is also continuous. lim
Conversely, if
(Hoo(V;F), II Ilv)
K then the
into (H(V;F),T ) is continuous and hence W oo (H (V;F), II II V) into lim (H(V;F) ,TW)
then for each
p
is a continuous seminorm on
V open,
V'::>K
there exists
c(V»
0
+ V~K
V open such that f
E
p (f) ~ c (V) II f II V
H(V;F)' Hoo(V;F)
then
Hence the restriction of
to
lim
(H(V;F),T ) w
+
f
in
Hoo(V;F).
If
and the same inequality holds.
Ilfllv p
ported by the compact subset norm on
for every H(V;F)
K of
V.
is a
T
Thus
P
w
continuous seminorm is also a continuous semi
and this shows that the two topologies coincide
V::) K
V open on
H(K;F)
and completes the proof.
It follows that
H(K;F)
is abornological space if
F
is a normed
linear space and an ultrabornological (and hence a barrelled) space if is a Banach space. then
H(K;F)
If
E
is a metrizable space and
F
F
is a Banach space
will be a countable inductive limit of Banach spaces and
hence a bornological
DFspace.
Thus we see that the space of germs will
always have some good topological properties since it is an inductive limit and indeed the main topological problems connected with
H(K;F)
are those
generally associated with inductive limits (as opposed to those connected with projective limits) such as completion, description of the continuous seminorms (sometimes we only need a description of sufficiently many
86
Chapter 2
continuous seminorms) and a characterization of the bounded sets. encounter all of these problems in later chapters. selves to characterizing bounded sets when Let
E
= lim Ea
We shall
Here we confine our
E is metrizable.
be an inductive limit of locaUy conVex spaces.
The
+
a
inductive limit is said to be regular if each bounded subset of contained and bounded in some
B
E. a
K be a compact subset of a locally convex space W be a convex balanced open subset of E. Then
Lemma 2.54 let
E is
{f£H(K+W); IlfllK+w
Proof
E and
Let
Let
{fa}aEA
is a cLosed subset of H(K).
(I}
be a convergent net in
is a Cauchy net in
H(K)
which lies in
show that
{fa}a£A
subset of
K+W
(H(K+W),T O) '
L C K+pW.
By using the Cauchy inequalities we see that
then there exists a real number
~~~ I am:~)
If
p,
B.
L is a compact
O
such that
(YXli
yd defines a continuous seminorm on
H(K)
sup x£K yd
+
for every nonnegative integer Since
and also on
{fa}aEA
as
0
(H(K+W),T
a,S
+ co
m.
<:B the Cauchy inequalities imply that I
lim sup m ~oo Hence, for some II f f II ~ a S K+L
sup xEK yd
[
sup aEA,x£K
c>O
I:=o
and all
sup xEK yd
1
i1iT
I'm
dm:~(Xl 11:1 '
II
1
a, SEA, "m
1 f ) (x) (vx) iii! d (fa s'
d (f f )(x)(yx) a B
+
N+l c_p__ Ip
and
O
)'
Hence
We
Holomorphic mappings between locally convex spaces this
tends to zero as
a+oo
in
H(K).
a,8
+
since
00
By the above, there exists a
uniformly on the compact subsets of
g
K+W.
dmf(~) = ~g(~)
for all
If
convex space then Proof
Since
in
f
+
B such that
as fa
+
g
Pm , K, L(f) = Pm , K, L(g)
and all
H(K)
fa
L of
K+W
it
K.
Hence
f
in
for and
and this completes the proof.
K is a compact subset of a metrizable locally
is a regular inductive limit.
H(K) E
g
Since
m
define the same equivalence class in
Proposition 2.55
Now suppose
m and every compact subset
every nonnegative integer follows that
O
87
is metrizable it contains a countable fundamental
neighbourhood system at zero,
(Vn)n'
and hence
H(K) = lim (H (K+Vn ),II lin) OO
+
n
is a bornological
DF
space.
Hence each bounded subset of
contained in the closure of a bounded subset of some
H(K)
Hoo(K+V).
Since the
n
closed unit ball of
Hoo(K+V ) is also a closed subset of n 2.54, this completes the proof
is
H(K), by lemma
A seminorm characterization of the bounded subsets of
H(K)
is given
in the following proposition
Let
Proposition 2.56
K be a compact subset of a metrizable locally
E. A subset B Of H(K) is bounded if and only if it satisfies each of the following conditions:
conVex space
(aJ
for each continuous seminorm
p
on
H(O)
~
sup p ( d f(x) ) XEK n!
(bJ
<
(xn)~=l and (x~)~=l are two convergent sequences in K. (Yn)~=l and (Y~)~=l are null sequences in E. (kn)~=l is
if
a strictly increasing sequence of positive integers and xn+Yn = x~+Y~
for all
then
n
,... sup fEB
Proof
L:=l
k 2 n
k
Lj~O
dJf(x ) n
.,
J .
kn (Yn )  Lj=o
djf(x') n j !
We first show that the seminorms given in
(*)
(Y~)
and
<
(** )
00
(**)
are
88
Chapter 2
continuous with respect to the inductive limit topology
on
HCK).
Using the above notation we let
1::=0
PI (f)
sup xEK
p C
and
k
k
1:~=o z n
pZ (f)
for every
f
Since and n
H(O)
in
dnf nl
1'\
n
Lm=o
dmf(x ) n (Yn) m!
k
/'
dmf(x' ) n
n
 Lm=o
(Y~) I
m!
HCK).
(H(K),T)
is a barrelled locally convex space and both
induce the
T
~ (nE)
topology on
w
H(K)
for each positive integer
it suffices to show each of the above seminorms is finite. Let
such that
f s H(K) f
be arbitrary.
There exists a neighbourhood ro
can be identified with an element of
H (K+4V).
V of zero Let
M = sup ;f(x+4y)I . xEK,ysV Since
p
Cry) > 0
Pl(g}
is a continuous seminorm on such that
L~=o
peg)
sup XEK
~
I'n p ( d f(x) ) nl
and each seminorm of the form
n>N.
(HeO),T)
C(V) Ilgll v
(*)
Now choose a positive integer For all n>N
there exists a constant
for every
~
C(V}
g
in
1:=0
4n
M
H(O).
Hence
<
is continuous. N
such that
Yn
and
y' n
E
V for all
89
Holomorphic mappings between locally convex spaces
and hence
2M
k
2
M
n
<
M
k
ce.
4 n
Thus each seminorm of the form set of
H(K)
is continuous and any bounded sub
(**)
satisfies conditions (a) and (b).
Conversely, suppose (a) and (b).
(f) is a subset of H(K) which satisfies a aEr Using seminorms of the form (*) we see that /'
dnf (x) a
nl
aE:,XEK,
is a bounded subset of
H(O).
n arbitrary
Since
there exists a neighbourhood of
0
H(O) in
is regular (proposition 2.55)
E,W,
and
M>O
such that
I dn:~ (x) II. ' for every If
a XEK,
in
f,
yEW
x and
in aEf
K and all
n.
let
(y).
nl To complete the proof, it suffices to show that there exists a neighbourbood
V of zero in
E,
x,x' E K, y,y' E V with
V C W, x+y
such that
= x'+y'
fa (x)(y)
and every
a in
=
fa (x')(y') f.
for all
90
Chapter 2
If not, there exist E,
(Yn)~=l
xn+Yn =
x~+y~
sequences in such that
two sequences in
(Y~)~=l'
and
for all
f
(xn)~=l
K,
(x~) ~=l'
and
(an)~=l
and a sequence
two null in
[
nand
(y') \ a(x') nn n
Now choose inductivel
(kn)~=l
such that
ka
strictly increasing sequence of positive integers
2 nOn> n+2M
for all
n.
Let q(f)
Since
1:=0
(fa)aEf
k 2 n
.\
k n 1m=o
" (x ) dmf n m!
I
(Y n )
satisfies condition (b),
I'
k
dmf(x' ) n
n m=O
sup q(fa)
(Y~)
m!
<
00
a
On the other hand,
m!
k
2 no
n
 2M > n.
This is a contradiction, and completes the proof. Condition (a) says that bounded subsets of estimates on
H(K)
satisfy Cauchy
K while condition (b) says that the Taylor series expansions
satisfy certain coherence properties.
If
K satisfies certain connected
ness conditions, then condition (a) is sufficient.
That some conditions on
K are necessary for boundedness is shown by the following example.
a:. For each positive integer n let 1 _l_} x+iy, x < and let IV {zE(;;z=x+iy,x >   1 }. 1 n n+ n + Z Z
Example 2.57 V = {z E (;; z n Let
fn E H(V n
U
Let
E
Wn )
be identically one on
Vn
and identically zero on Wn
91
Holomorphic mappings between locally convex spaces If
K=
I {n} n:;:l
for all
n.
u {O}
then
K is a compact subset of
{fn}~=l
The sequence
and
([
fn
E:
H(K)
satisfies condition (a) of proposition
2.56 but is not bounded by proposition 2.55. We shall use proposition 2.56 and its method of proof again in a later chapter.
The following result follows easily from the corresponding result
for holomorphic functions on open sets.
If
Proposition 2.58 E and
K is a compact subset of a locally convex space
is a complete locally convex space, then
F
closed complemented subspace of H(K;F)
for every positive integer
In discussing holomorphic functions on a compact subset locally convex space complete.
E we may always suppose that the space
"E
If not, let
is a
(@ (mE;F),T) w
denote the completion of
E.
If
m.
K of a E
is
V is a convex
balanced open subset of E then the natural restriction from Hoo(K+V) oo into H (K+V () E) is a bij ecti ve isometry and hence the spaces H(K ) and E H(Kg) are isomorphic as locally convex spaces, (KE(respectively Kg) is the set
K considered as a subset of
E
(respectively
E)).
We shall also need hypoanalytic germs in later chapters and the definitions are exactly as one would expect.
If
K is a compact subset of a locally convex space
locally convex space then on ence relationship on which both
f
~
and
by g
U
E and
F is a
HHy(V;F) l,)e define the equivalKCV, V open f'" g if there exis ts a neighbourhood W 0 f K
are defined and coincide.
We let
U ~y(V;Z'F) Kev
.rvI
V open
and the elements of this vector space are called Fvalued hypoanalytic germs on
K.
On
~y(K;F)
we define the locally convex inductive limit topology

lim
V:;>K , V open
We have used an inductive limit of topology on
H(K;F).
TW
topologies to define the natural
It is not known if the
as a projective limit of the spaces
H(K;F).
TW
topology can be recovered
This problem is rather impor
tant and appears to be intimately related to the localization problem
Chapter 2
92
mentioned previously:
is
Tw
a local topology?
inquiry, we can define a "new" topology on a locally convex space and Tn·
It is conjectured that
If
Definition 2.59
and
F
H(U;F),
Tn
and
TW
always coincide.
U is an open subset of a locally convex space
F is a locally convex space then the
H(K;F)
One easily sees that TO:i Tn:i TW on give examples of situations in which T n
2.60
Show that
topology on
Tn
E
is
H(U;F)
K ranges over the
where
u.
compact subsets of
EXERCISES
U an open subset of
a locally convex space, which we denote by
defined as the projective limit of
§2.5
Following this line of
In chapter six, we shall coincide.
T
W
for any open subset
H(U;F)
HG(U;F)
dimensional space
H(U;F). and
E and any locally convex space
F
U of an infinite
if and only if
E ,:::: «(N) . 2.61*
(E,t f )
Show that
is a topological vector space if and only if
E
has a countable algebraic basis. 2.62 E onto
If
E and
F
F are vector spaces and
for any 2.64*
If
x Let
E. Let f for every 2.65*
is a linear mapping from
is an open mapping.
+
2.63
TI
show that
E and
F are vector spaces and
in
Yl' ... 'Yn
E,
E E
Let
and
E and
Show that
f
£
£
Jl~(mE;F)
show that
n:im.
F be Banach spaces and let
HG(U;Fti). in F. E and
£
A
H(U;Fti)
F be Banach spaces and
U be an open subset of i f and only if
0
f
E
H(U)
U a connected open subset
93
Holomorphic mappings between locally convex spaces
of
E.
Let
fey) C F' 2.66
f: U .. F*
E.
If feU)
E and
f E: H(U;F)
0
f E: H(U)
for every
V of
U.
F be Banach spaces and let
Let
in
F
and
f E: H(U,; Fil)·
U be an open subset of
show that the closed vector subspace of
E and
cp
Show that
is equal to the closed vector subspace of
u 2.67*
cP
for some nonempty open subset Let
by
and suppose
F
generated
F generated by
I'
m,~E:U,xE:E dmf(~)(x)
F be Banach spaces.
to be compact at the point
x
A mapping
f: E..F
is said
if there exists a neighbourhood
such that
f(V x ) is a relatively compact subset of show that the following are equivalent:
F.
If
Vx
of
x
f E: H(E;F)
is compact at every point of E, is compact at some point of E,
(a)
f
(b)
f
(c)
there exists a compact subset feE)
K
of
F
such that
is contained in the vector space spanned by
K, the transpose mapping
(d)
(where
f*(cp) = cpof)
1\
(e)
dnf(O)
f*;(F'" o ) .. (H(E),T 0 ) is continuous,
is compact for all
n.
E where each is a locally convex space. Ln=l n Suppose fn E: H(En) for each n and let gn E: ~ (I~=l Ek ) for each n. If there exists a neighbourhood U of zero in E such that iignii < + 00
2.68*
Let
for all
n
2.69
Show that, the composition of holomorphic (respectively
E
,00
show that
F 
rn=l gn f nE:
u
H(E).
Mholomorphic, hypoanalytic) functions is holomorphic (respectively Mholomorphic, hypoanalytic). 2.70
If
f
is a locally bounded holomorphic function,
continuous holomorphic function and
gof
g
is a Mackey
is defined, show that
gof
is a
(continuous) holomorphic function. 2.71
Let
U be an open subset of a locally convex space
be a locally convex space.
If
f E: HG(U;F)
and either
E or
E and let F
is
F
94
Chapter 2
separable, show that 2.72*
E ~y(U;F)
f
if
gof
E
H(U)
f
HG(E)
If
E
is a Frechet space and
and only if
f
is a Borel measurable function.
2.73*
Let
then
f(x)
(en)~=l.
I:=l fn(x)e n
The mapping of
f
If f
where
f
g
show that
U be an open subset of a Banach space
a Banach space with a basis, F
E
for every f
in
E
H(E)
E and let
is a mapping from
H(F).
F
if
be into
U
:U+('
n
is said to be normal if, for each compact subset
U and each positive
6,
there exists a positive integer
n(K)
K such
that for all If
fn
E
H(U)
for all
n
show that
f
E
m> n(K) .
H(U;F)
if and only if
f
is a
normal mapping. 2.74* in
Let
(~n)n
E be a locally convex space and let
be a sequence
E'. (a)
If
is a Frechet space or a ~:fm. space show that
E
I:=l( ~nf
H(E)
E
for every
x
~n (x)
i f and only i f
in
+ 0
as
n + 00
E. oo
(b)
If
E = (E,a(E,E'))
only if of 2.75*
If the locally convex space
is
such that
if
EB
E
E,
dnf(x)
F E
(the vector span of
B in
a?>
is a Banach space and CPM(nE;F)
for some
x
f
E
in
TM
HG(U;F)
If
f
TM
complete locally convex spaces, and
HG (U; F)
continuous show that
f
where
E ~(U;F).
U is a
E
is
1M
complete.
complete locally convex show that
f
E ~(U;F)
U and every positive integer
2.76*
E
B normed by the gauge
then we say that
F
being
if and
contains a fundamental system of
U is a connected T M open subset of the
space
H(E)
is finite dimensional.
(~n)n
bounded sets
E
is a null sequence and the vector span
(~n)n
of B) is complete for every If
I n=l(~ n )n
show that
TM
open subset of f
E x F, E
n. and
is separately Mackey
95
Holomorphic mappings between locally convex spaces
2.77
If
U is an open subset of a locally convex space
locally convex space, show that for each
f
in
HG(U;F)
E and
F
is a
the following are
equivalent: (a)
f E H(U;F)
(b)
for each
(respectively in
E;
U
,II
dmf(E,) )"" m=O m! is an equicontinuous (respectively locally bounded) family of mappings. 2.78
E be a Banach space and let
Let
sequences in
(H(E) "0) be defined by L is well defined and holomorphic.
Show that If
f
E
be the space of all null
endowed with the sup norm topology.
E'
L : Co (E ')
Let
2.79
Co (E')
>
H( IT E)
where each
is a locally convex space, show
of
A and
a
Al
such that
.J
f
= fOIT
If
"" n Ln=l (
E
aEA a
that there exists a finite subset
L( (
where ITA is the natural projection from Al 1 is metrizable for each a in A show that E
IT E aEA a
onto
a
H_y(ITE)
11
2.80*
If
and,kS(y),
X and
H( ITAE ). aE
aEA a
a
Yare completely regular Hausdorff spaces
and .Ra (X)
each with the compact open topology, are infrabarrelled, show
that
2.81
Show that
and any open subset
H(U;F) U of
HHy(U;F)
Jo
(X)
where
for any locally convex space
F
X is a paracompact topological
space. 2.82
Show that
any open subset
H(U;F)
= ~(U;F)
U of £, (X)
where
for any locally convex space X is a Lindelof space.
F
and
96
Chapter 2
2.83*
If
r
is an uncountable discrete set and
F
is a separable
Banach space, show that H(c
H(co(r);F) where the topology on
co(r) c
o,p
(r);F)
is given by the sup norm and
(r)
o,p
r 'e r r' countable 2.84* F
u be a
Let
open subset of a locally convex space and let
be a locally convex space.
Let
denote the topology on
~(U;F)
of uniform convergence on the finite dimensional compact subsets of
U.
Show (a)
(~(U;F)
,IF)
is metrizable i f and only if
E
contains
a countable fundamental system of finite dimensional compact sets and (b)
Show that ~(U;F)
(c)
(d)
if If
2.85*
(H([I),,) o
If
2.86
subsets of 2.87*
the U
is a semiMontel space if and
is a semiMontel space. (~(U;F); 'F)
is quasicomplete i f and only
is quasicomplete. I
is an arbitrary set, show that the bounded subsets of
are locally bounded. E
is a complete locally convex space and the
H(E)
Let and
'0
of
F
(~(U;F)"F)
F
Show that
is metrizable.
'F induces on each 'F bounded subset of the topology of pointwise convergence.
Show that only if
F
are locally bounded, show that
E
'0
bounded
is barrelled.
E = I~=l En where each En is a Banach space. Show that bounded subsets of H(U) coincide for every open subset
E.
topological space
= (Fn)~=l of a is closed and each com
pact subset of
U be an open subset of
2.88*
We say that a countable increasing cover U is kdominating if each Fn U is contained in some Fn. Let
'1
97
Holomorphic mappings between locally convex spaces
a locally convex space all
H J (U)
n}.
E and let
{fE:HG(U); Ilfllp<
Let
lim H J (U)
(H (U) , TO)
n
where
J
J.T
K
ranges over all countable increasing kdominating covers of To
for
00
is endowed with the locally convex topology lenerated by Pn (f) = II ~~
the seminorms
H;1(U)
is the bornological topology associated with
TO
on
U.
Show that
H(U).
Show that
K
(H(U),T
O
)
is an ultrabornological space if
U is a kspace.
K
2.89
On
H(U) ,
U an open subset of a locally convex space, let 00
T P
If J = (Pn)n=l is a U consisting of closed sets and H J (U) the associated metrizable space of holomorphic functions, show that
denote the topology of pointwise convergence. countable increasing cover of 1 im H J (U)
is the bornological space associated with
(H (U) ,Tp) .
is
( if
T ranges over all possible covers of the above type). 2.90 and
If P
U is an open subset of a metrizable locally convex space
is a Banach space, show that
(H(U;P),T O)
E
is a regular inductive
limit of Prechet spaces. 2.91*
If the locally convex space
convergent sequence, show that
E'
13
E contains a nontrivial very strongly contains a sequence which is not very
weakly convergent. 2.92
If
E
is a separable locally convex space in which every sequence
is very weakly convergent, show that
E'
2.93 ent:
Show that the following are equival~
Let
E be a Prechet space.
13
admits a continuous norm.
(a)
E contains a nontrivial very strongly convergent sequence,
(b)
E contains a subspace isomorphic to
(c)
[(N)
(d)
is a quotient of E' 13' every continuous seminorm on
~N,
E has an infinite dimensional
kernel. Show that (a) and (d) are not equivalent for arbitrary locally convex spaces.
98
Chapter 2
Let
2.94*
{x£~L
:3 (xn )00n= 1
convergent sequence}. f £ H(E),
A
E be a metrizable locally convex space with completion £ E
such that
Show that
(xnx)n
E.
is a very strongly
is a vector subspace of
"E.
If
show, by using the factorization lemma or otherwise, that there
1
exists a unique
in
H(E*)
such that
ftE
=
f.
Generalise this result
to arbitrary locally convex spaces. 2.95
Let
(H(U)"o)
U be an open subset of a locally convex space.
is a locally mconvex algebra.
multiplicative seminorm on 2.96*
Let
H(U)
Let
f £ H(U;F).
Fvalued holomorphic germs at roJ
{germ of
f(x)
1
2.97*
If
TO
Show that every
£
If
in
0
f
f: E)
at
Tw
continuous
continuous.
U be an open subset of a Banach space
a Banach space.
show that
is
Show that
E
U + H(O;F)
and let
F
be
(the space of
is defined by x
translated to the origin}
HG(U;H(O;F)).
K is a compact subset of a locally convex space
is a locally convex space, show that
H(K;F)
E
and
F
is a Hausdorff locally convex
space. 2.98*
Show that
H(K)
K is a
is never a strict inductive limit when
compact subset of a locally convex space. 2.99* space
If
E
K is a convex balanced subset of a metrizable locally convex
show that
neighbourhood
V
Be H(K)
of zero and two positive numbers
11~llv~ for every 2.100*
t;
If
in
is bounded if and only if there exist and
c
C
2.101*
F'
s
~
If
such that
c . em
K and every nonnegative integer
m.
U is an open subset of a Banach space, show that
a dual Banach space (i.e. show that there exists a Banach space that
a
Hoo(U) F
is
such
Hoo(U)). K is a compact subset of a metrizable locally convex space,
99
Holomorphic mappings between locally convex spaces
show that
H(K)
is a locally
m  convex algebra.
Show that a convex balanced subset of
2.102
of a locally convex space
E,
is a
if it absorbs every equicontinuous subset of Let
2.103* integer
n
Tn
U an open subset
H(U).
E be a locally convex space.
let
H(U),
neighbourhood of zero if and only
denote the topology on
For each nonnegative H(E)
generated by the semi
norms
where
K ranges over the compact subsets of
bounded subsets of Show that
E
and
B ranges over the
E.
= Tn+l
Tn
each bounded subset of
E
for some (and hence for all)
n
if and only if
is contained in the closed convex hull of a
compact set. 2.104*
A subset
K of a locally convex space
E
is said to be strictly
compact if there exists a convex balanced bounded subset
B of
E
such
E . On ~(E) let denote the B topology of uniform convergence on the strictly compact subsets of E.
that
K is contained and compact in
Show that
(~(E)
2.105*
If
f
,TS)
is a semiHontel space.
is an entire function on the J)J'h1 space
there exists a neighbourhood every positive integer
§2.6
V of zero such that
Ilf I~v
E,
show that
is finite for
n.
NOTES AND REMARKS The historical notes of A.E. Taylor [680] are probably the most
comprehensive guide to the development of infinite dimensional holomorphy up the mid'nineteen forties available and we have made extensive use of these notes in §1.6 and shall do so again in this section.
In another
paper, [681], A.E. Taylor documents the role of analyticity in operator and spectral theory and his history of the differential in the nineteenth
Chapter 2
100
and twentieth centuries [682] contains much of interest concerning the growth of infinite dimensional analysis.
E. Hille and R.S. Phillips ([334],
section 3.16 and chapter 26) also provide a summary of the fundamentals known at that time  1958  together with a short bibliography on the subject (chapter 26 was written in collaboration with M.A. Zorn).
D. Pisanelli
[578] gives a survey of most of the different concepts of holomorphic function that are currently in use and documents their origin. V.I. Averbukh and O.G. Smolyanov ([40],§2) give a comprehensive account of the development of the concept of the vector spaces.
differ~ntial
in topologicru
They are mainly concerned with the real theory but natur
ally this has many consequences for the complex case.
An interesting point
that emerges from [40] is that the search for a definition of differentiable function between real linear topological vector spaces has been in progress more or less continuously for the last seventy years with many different definitions being proposed  the authors list twentyfive in [40]  and since a number of definitions were not seen to be equivalent and indeed, various authors were not aware of one another'S work, this led to a certain a mount of chaos  or at least apparent chaos  which has only been rectified in [40].
Fortunately for differentiation over complex spaces we
have power series expansions and this has led to a fairly unanimous acceptance of Frfchet differentiability as the standard definition with minor roles being played by G~teaux holomorphic, Mackey or Silva holomorphic and hypoanalytic functions. The names chiefly associated with the development of real infinite dimensional differential calculus are (see [40] for more detailed information) V. Volterra, J. Hadamard, M. Frechet, R. G~teaux, P. Levy, A.D. Michal, L.M. Graves, E.W. Paxson, J. Gil de Lamadrid, J. Sebasti'ao e Silva, H.H. Keller and G. Marinescu.
In particular, we would like to mention
J. Hadamard, M. Frechet and A.D. Michal.
Hadamard was one of the first to
recognise the importance of Volterra's work, he exercised a deep influence on his students Frechet and G~teaux and was always very insistant on linearity being incorporated into any definition of the derivative. M. Frechet and A.D. Michal both spent a considerable portion of their lives developing the theory  roughly forty years by Frechet  and opened many avenues still being explored.
Readable accounts of the real theory are to
be found in P. Levy ([442], both the 1922 and 1950 editions), A.D. Michal
101
Holomorphic mappings between locally convex spaces
[491], V. I. Averbukh and O.G. Smolyanov [39], J.P. Penot [567], A. Frohlicher and W. Bucher [247] and S. Yamamuro [718]. Surveys of a more specialised nature but which are more relevant to the topics discussed in this book are L. Nachbin [518], M. Schottenloher
[642], K.D. Bierstedt and R. Meise [70], P. Kr~e [405] and S. Dineen [198]. We now present our brief history. We have already noted in §1.6 that the definitive steps in the creation of real and complex infinite dimensional differential calculus were taken by V. Volterra (1887) and D. Hilbert (1909) respectively. Prior to these, the works of J. Bernoulli on the curve of quickest descent, L. Euler on the calculus of variations (see J. Hadamard [298]) and H. Von Koch [393], on infinite systems of differential equations may also be mentioned as of particular importance for later developments.
D. Hilbert
[332] discussed analytic continuation and the composition of holomorphic functions but his work is not easy to read, and at times rather vague. Next came R. G~teaux [251,252,253] who looked at both real and complex analysis  as opposed to M. Frechet who only studied the real case  and set out to clarify and extend the works of Frechet and Hilbert. he succeeded brilliantly.
In this
In [252], he defined complex polynomials and
noticed their relationship with bilinear and quadratic forms, defined the "G~teaux" derivative, extended the Cauchy integral formula and the Cauchy
inequalities, proved an identity theorem and various convergence theorems for holomorphic functions, established the now standard correspondence between derivatives of a holomorphic function and the homogeneous polynomials in its Taylor series expansion and gave various results about analytic continuation and power series expansions  all in an infinite dimensional setting.
His other paper [253] is also quite interesting and
contains brief outlines of a number of topics which his untimely death prevented him from developing.
II:N,~j [a,b]
and
G~teaux's work was confined to the spaces
R. •
2
In 1920, S. Banach gave, in his thesis, the axioms for a complete normed linear space (a Banach space) and in 1923, N.Wiener
[715], who
incidentally gave independently the same axioms three months after Banach,
Chapter 2
102
noticed that the Cauchy integral formula extends to Banach valued holomorphic functions of one complex variable and in this setting, many of the classical results such as Morera's theorem, Abel's theorem and the residue theorem are also valid.
L. Fantappi~'s long article [234] appeared in 1930.
In it he proposed that a continuous function
f
aBanach space be called holomorphic if
is holomorphic (as a function
f
0
defined on a domain
of one complex variable) for any holomorphic function the complex plane into
D.
D in
from a domain in
Fantappie's work served as motivation for A.E.
Taylor [675] and was developed and put in a modern context using topological vector spaces by J. Sebasti'aO e Silva [649,650,651,652,653] (see also D. Pisanelli [578]). In the nineteen thirties, an intensive study of the whole field of analysis and geometry in abstract spaces was carried out by A.D. Michal and his students R.S. Martin, A.H. Clifford, I.E. Highberg and A.E. Taylor.
In
1932, R.S. Martin [449] developed the theory of holomorphic mappings for Banach spaces using a power series approach (see also A.D. Michal and A.H. Clifford [492]).
The final step towards the current definition of holomor
phic mapping between Banach spaces was taken independently by L.M. Graves in 1935 [278] and A.E. Taylor in 1937 [675].
Graves' treatment is rather
brief, p.649653, and in it he notes that Gateaux differentiability plus continuity are equivalent to Frechet differentiability (defined for normed linear spaces by Frechet in 1925, [245]) when dealing with functions between complex normed linear spaces and also that this leads to a satisfactory theory of holomorphic functions between normed linear spaces. Taylor's work is much more explicit.
He defines a holomorphic function as
a continuous function whose one dimensional sections are holomorphic.
This
definition is a natural outgrowth of the work of G~teaux and brings together the ideas of many of his predecessors.
He proved the following
result: If
f
is a mapping between normed linear spaces, then the following
are equivalent a)
f
is continuous and has a G~teaux derivative at each
point; b)
f
has a Fr~chet derivative at each point;
c)
f
has a power series expansion which converges
103
Holomorphic mappings between locally convex spaces
uniformly in a neighbourhood of each point. Taylor goes on to generalise Riemann's theorem on removable singulari ties, Mi ttagLeffler's theorem, Liouvi lle' s theorem and the CauchyRiemann equations.
He also shows [675,p.292] an awareness of the fact that
the radius of uniform convergence is not equivalent to the radius of analyticity and this is the first step in an extensive study of the radius of boundedness which we undertake in chapter 4.
In a further paper [678],
A.E. Taylor discusses a number of interesting examples of holomorphic functions on
~p
spaces and gives a generalization of Hartogs' theorem on
separate analyticity.
The Polish school of functional analysis (S. Banach,
S. Mazur, W. Orlicz, etc.) also contributed during this period (see chapter 1) and although this traditional interest in nonlinear functional analysis was maintained until relatively recently (see
for instance
A. Alexiewicz and W. Orlicz [7], W. Bogdanowicz [76] and A. Pelczynski [563,564,565]), it was largely overshadowed by the rapid development of the linear theory.
This predominance of linear functional analysis may be
traced to the influence of S. Banach's book [44], published in 1932, although Banach himself hoped to develop the nonlinear theory if we are to judge from his comment [44,p.23l] on complex vector spaces "Ces espaces constituent le point de depart de la theorie des operations lineaires complexes et d'une classe, encore plus vaste, des ope~ations analytiques, qui presentent une generalisation des fonctions analytiques ordinaires (cf. p. ex. L. Fantappie, I. funzionali analitici, Citta di Castello 1930). Nous nous proposons d'en exposer la theorie dans un autre volume". See also the preface to [44]. M.A. Zorn [723,724,725] made a number of important contributions in the midforties and we refer to his results later in the text.
The nine
teen fifties saw the appearance of A. Grothendieck's memoir [287] on topological tensor products and nuclear spaces and this has had and will continue to have a considerable influence on the development of infinite dimensional holomorphy.
During this period, J. Sebastia5 e Silva also
developed his theory of infinite dimensional holomorphy and H.J. Bremermann [103,104,105] proved various results on pseudoconvex domains and tubular
Chapter 2
104
domains of holomorphy in Banach spaces. This brings us to the modern period, the last sixteen years, during which time most of the work we discuss has been discovered.
The con
tributions of the various authors during this period are detailed in the final section of each chapter. The concept of Mackey holomorphic function is usually attributed to J. Sebastiao e Silva [652], although an earlier equivalent definition (condition 2. 18 (d) ) is due to L. Fantappie [234].
The theory of Mackey holo
morphic functions finds its most natural expression within the language of bornologies  see for instance D. Pisanelli [578], D. Lazet [423] and J.F. Colombeau [141]. E. Hille [333].
Locally bounded holomorphic functions were introduced by It is difficult to locate the origin of hypoanaliticity
(definition 2.15) but the name usually associated with the real analoguehypocontinuity is J.L. Kelley who also introduced the concept of kspace (see, for instance, S. Willard [716] or Z. Semadeni [654]).
Examples 2.16,
2.19 and 2.20 together with further examples of a similar kind are given in S. Dineen [190] (see also chapter 5).
The connection between Hartogs'
theorem and locally bounded holomorphic functions (proposition 2.12, examples 2.13 and 2.14) appears in [190] and infinite dimensional generalizations of Hartogs' theorem are discussed in the comments on exercise
2.7~
Factorization results (most of which depend on Liouville's theorem concerning bounded entire functions) are implicit in A. Hirschowitz [335], C.E. Rickart [60S] and L. Nachbin [514]. S. Dineen [188,189,190,191] and E. Ligocka [443], developed independently a general theory of factorization which we discuss in more detail in §6.2 and applied it to such topics as the Levi problem, Runges theorem, Hartogs' theorem, holomorphic completion, Zorn's theorem and topologies on spaces of holomorphic functions.
Factor
ization method$ have also been applied to the Levi problem by Ph. Noverraz [540,544,546], S. Dineen [183,186], B. Josefson [358], M. Schottenloher [640], J.F. Colombeau and J. Mujica [154], J. Mujica [506]) to the construction of the envelope of holomorphy by M. Schottenloher [640] and P. Berner [59,60], in studying meromorphic functions and the Cousin I problem by V. Aurich
[35] and S. Dineen [192], to the theory of convolution operators
by P. Berner [62] and M.e. Matos [463,467] and in studying locally convex topologies on spaces of holomorphic functions by P. Berner [61], S. Dineen
105
Holomorphic mappings between locally convex spaces
[194], and Ph. Noverraz [552].
Theorem 2.21 and proposition 2.33 are given in [190].
Example 2.22 and
proposition 2.24 are due to L. Nachbin [514] and a generalization can be found in S. Dineen [190]. subset U in
[N
factorization.
and
f
A. Hirschowitz [335] gives an example of an open
£
H(U)
which admits a local but not a global
This example shows that condition (*) of theorem 2.21 is
not sufficient to obtain a global factorization result.
The difficulties
posed by Hirschowitz's example were overcome by P. Berner [59] and M. Schottenloher [640].
Examples 2.22, 2.25 and 2.26 are due to L. Nachbin
[514], and the proof of example 2.22 appears in [193].
Proposition 2.27 is
due to S. Dineen [190]. Theorem 2.28 is due to M.A. Zorn [724].
Generalizations to other
locally convex spaces are due to Ph. Noverraz [536,537], D. Pisanelli [578], A. Hirschowitz [341], D. Lazet [423], J.F. Colombeau [141], E. Ligocka and S. Dineen [190].
[443]
A locally convex space
E for which the con
clusion of proposition 2.29 is valid is called an wspace (S. Dineen [190]).
[184],
E. Grusell [292,293] and M. Schottenloher [645] give examples of
locally convex spaces which are not wspaces.
Corollary 2.30 is due to
J.F. Colombeau and J. Mujica [154], who use it together with the solution to the Levi problem for Hilbert spaces CL. Gruman [290]) to solve the Levi problem in strong duals of Frechet nuclear spaces
(JJ5fL spaces).
Example 2.31, which arose in constructing a counterexample to the Levi problem, is due to B. Josefson [358] and the proof given here is taken from S. Dineen [193].
The inequality of example 2.31 is used in A. Renaud
([604, proposition 4) to obtain a vectorvalued Schwarz lemma.
Other
factorization results may be proved by considering locally bounded holomorphic functions and in such cases one gets results for nonnormed range spaces [190]. The topological vector space structure of [n,
H(U),
U an open subset of
has been studied by a number of authors including A. Grothendieck
[285], G. Kothe [396]
and A. Martineau [451].
The compact open topology
was first investigated on spaces of holomorphic functions in infinitely many variables by H. Alexander [5] and L. Nachbin [509].
They found, how
ever, that it did not possess very good topological properties and so, motivated by properties of analytic functionals in several complex variables
106
Chapter 2
due to A. Martineau [450], L. Nachbin [509] introduced the 1w (definitions 2.33 and 2.34).
The
10
topology
topology (definitions 2.33 and 2.34)
was first introduced for holomorphic functions on separable Banach spaces by G. Coeure [128,129] in connection with problems of analytic continuation and the general definition is due to L. Nachbin [510]. and 2.41 are given in S. Dineen [185].
Propositions 2.38
Lemma 2.39 appears to be well
known and a proof is given in KD. Bierstedt and R. Meise [69]. corresponding problems for the
1w
and
18
The
topologies, i.e. are these
local or sheaf topologies? are perhaps the most interesting current open problems in the general theory of locally convex spaces of holomorphic functions in infinitely many variables.
A related open problem is to
characterize the open sets in which 10
and
10
continuous multiplicative
linear functionals are point evaluations (by exercise 2.95 the the
1
W
1
and
o
continuous multiplicative linear functionals coincide).
This
problem arises in solving the Levi problem and in constructing the envelope of holomorphy.
Results dealing with this problem are to be found in
H. Alexander [5], G. Coeure [129,131], M. Schotten10her [632,633,634,635, 641,643], J.M. Isidro [352] and J. Mujica [502,503,504,505].
Lemma 2.43,
corollary 2.44 and example 2.46 are proved in S. Dineen [185]. 2.45 and example 2.47 are to be found in S. Dineen [194].
Corollary
For;JJ:1;"
(strong duals of Frechet Schwartz) spaces, example 2.47 is due to J.A. Barroso, M.C. Matos and L. Nachbin [50].
Reference [194] is devoted to the
theory of ho10morphic functions on .:tYf111 (strong duals of Frechet Montel spaces) and it is interesting to note that although many results on ~1~ spaces (for instance, example 2.47) can be extended to holomorphic functions on i)}Yrispaces, the methods of proof are quite different.
On Jj jJ'
spaces, one uses frequently the property that such spaces are inductive limits of Banach spaces with compact linking mappings in the category of topological spaces (this is not true of JJ:Jh'l spaces) while on iJ:f
Yrz
spaces, one uses the fact that such spaces are kspaces and hereditary Lindelof spaces.
Example 2.48 is due to L. Nachbin [509] and example 2.49
can be found in S. Dineen [185]. Very strongly convergent sequences(definition 2.50) were introduced by S. Dineen [184] in studying holomorphic completions (see §4.4).
Subsequent
ly they were used by S. Dineen [185] in counterexample 2.52 and in [190] they were applied together with very strongly convergent nets and very weakly convergent sequences (definition 2.51) in other areas of infinite
107
Holomorphic mappings between locally convex spaces
dimensional holomorphy.
Further applications are to be found in P. Berner
[58,59] and S. Dineen and Ph. Noverraz [205].
J.A. Barroso [46] shows that
T = T on H((A) if and only if A is countable and this, together with o w example 2.52, completes the picture of the relationship between the differ
ent topologies on
H((A),
A arbitrary (see also chapter 5, J.A. Barroso
and L. Nachbin [53] and V. Aurich [33]). Holomorphic germs on compact subsets of Banach spaces were first investigated by S.B. Chae [119,120] and A. Hirschowitz [339,343]. quently, J.
~1uj
Subse
ica [S03] developed the theory on metrizable locally convex
spaces (see also R.R. Baldino [43], A.J.M. Wanderley [714], P. Aviles and J. Mujica [41], KD. Bierstedt and R. Meise [69,70] and E. Nelimarkka [525]) Lemma 2.54 and proposition 2.55 are due to J. Mujica [503].
Earlier,
S.B. Chae [120] and A. Hirschowitz [339] had given incomplete proofs for the Banach space case.
In providing a complete proof, A. Hirschowitz
introduced seminorms similar to those which appear in proposition 2.56 (see also S. Dineen [200]).
The question of whether or not condition (b)
of proposition 2.56 is redundant is answered negatively by example 2.57 (due to R.M. Aron) and has led to some interesting investigations by W.R. Zame [721], K. Rusak [617] and J.T. Rogers, Jr. and W.R. Zame [609] in finite dimensions and by R.L. Soraggi [664,665,666] in infinite dimensions. Rogers and Zame show that condition (b) is not necessary for compact subsets of
[
which have only a finite
nl~ber
of connected components and
this result does not extend to higher dimensional spaces.
For arbitrary
locally convex spaces, sufficient conditions for the removal of (b) are obtained by placing rather technical local connectedness conditions on the compact set. The
Tn
topology was introduced by S.B. Chae [120] and it has also
been studied by J. Mujica [503] and K.D. Bierstedt and R. Meise [70], (see chapter 6).
This Page Intentionally Left Blank
Chapter 3
HOLOMORPHIC FUNCTIONS ON BALANCED SETS
In many branches of function theory there are collections of sets which arise naturally and possess properties which ensure that they play an important role  for example convex sets in functional analysis and effile sets in potential theory.
In the theory of several complex variables many
different kinds of special sets arise  pseudoconvex sets, polar sets, polynomially convex sets, polydiscs and Stein manifolds. Balanced open sets arise as the natural domain of convergence of the Taylor series expansion at the origin for holomorphic functions.
In this chapter we consider all the
holomorphic functions on a balanced open set U and show that the Taylor series expansions lead to a topological decomposition of H(U) chapter 2.
for the different topologies we discussed in
We then use this decomposition to deduce top
ological properties of H(U)
and to extend results about
spaces of homogeneous polynomials to spaces of holomorphic functions. Schaude~
Our main tools are associated
topo~ogies
and
decompositions and since these topics are not very
well known we devote the first section to their exposition. The theory of holomorphic functions on balanced open sets may be regarded as a
~oca~
theo~y
that Tw and To were local
(or sheaf)
and if we could show
topologies then the
results of this chapter would extend to arbitrary open sets. 109
Chapter 3
110
§3.l. ASSOCIATED
TOPOLOGIES
DECOMPOSITIONS IN
AND
LOCALLY
GENERALISED CONVEX
SPACES
In linear functional analysis we encounter two mutually exclusive but not exhaustive types of properties
 those
preserved under locally convex inductive limits and those preserved under projective limits.
For example the locally
convex inductive limit of barrelled (respectively bornological) locally convex spaces is barrelled (respectively bornological) and the projective limit of complete 1 0 call y c on vex spa c e s i s com pIe t e
(respectively nuclear)
( res pee t i vel y n·u c 1 ear) .
Many other properties arise as a combination of properties from these two types,
for example reflexivity is defined
as semireflexivity (preserved under projective limits) plus infrabarrelledness limits)
(preserved under locally convex inductive
and the combined property is not preserved under
A general method used in showing that
either kind of limit. a locally convex space, associate with
(E,T)
(E,T), has a given property is to
another locally convex space
the given property and then to show that
(F,T')
with
(E,T) = (F,T').
For
example we can associate with a given locally convex space its completion and also its associated bornological space.
A
systematic theory has been developed for properties which are preserved under locally convex inductive limits and we now discuss this theory and apply it in later sections to the theory of hOlomorphic functions.
A Q family of locally convex topologies
Definition 3.1
3 of
is a family
locally convex topologies such that
(a) ~ is stable under locally convex inductive limits, (b)
on any vector space the finest locally convex topology
belongs to 1. Let (E,T) be a locally convex space and let
Q family of locally convex topologies.
1
be a
The locally conVex
inductive limit of all topologies on E which lie in ~ and are finer than
T
also lies in~.
and call it the
~
We denote this topology by
topology associated with
T.
T:
III
Holomorphic functions on balanced sets The topology T j
can also be characterized as the solution
to a universal problem. convex space and
~
Specifically if (E,T)
is a Qfamily then
T~
is a locally ~
is the unique
topology on E such that any continuous linear mapping from (F,T')
into E, T'E],
factors through
(E,T g ).
For certain Qfamilies there are also other known characterizations which can be useful. Example 3.2
Let b denote the family of all bornological
topologies.
Then b is a Qfamily and for any locally convex
topology, T,
we let Tb denote the associated bornological
topology.
If (E,T)
is a locally convex space and B is a
closed convex balanced bounded subset of E then we let EB denote the vector subspace of E generated by Band normed so that B is its closed unit ball. (E,T ) b
lim
"""l3"*
It is well known that EB
where B ranges over all closed convex balanced bounded subsets of E. Example 3.3
Let ub denote the family of all ultra
bornological locally convex topologies.
Then ub is a Qfamily.
For any locally convex space (E,T) we let Tub denote the associated ultrabornological topology on E.
We have,
as in
example 3.2, (E,T ub )
=
l~m)
EB
where B ranges over all complete balanced convex bounded subsets of E. Example 3.4
A locally convex space,
(E,T) is called a
Kelley space if any finer locally convex topology on E has less compact sets. topology.
An ultrabornological topology is a Kelley
The collection of all
locally convex Kelley
topologies is a Qfamily and we let TK denote the Kelley topology associated with the locally convex topology T.
112
Chapter 3 The barrelled and infrabarrelled topologies also form
Qfamilies and for a given locally convex space
(E,,) we let
' t and ' i denote the associated barrelled (tonnele) infrabarrelled topologies respectively.
and
An alternative
description of these topologies, by means of transfinite induction, is given in proposition 3.5.
The collection of all
barrelled and bornological locally convex topologies, bt,
is
also a Qfamily and we denote by 'bt the bornological and barrelled topology associated with " For any Hausdorff locally convex space
(E,,) we have the
following inclusions between the various associated topologies, where a
~
b means that a is finer than b,
We complete our discussion of associated topologies by showing that the barrelled topology associated with a complete topology is also complete. Proposition 3.5
If (E,,) is a complete locally convex space
then (E"t) is also a complete locally convex space. Proof
We first construct an ordered family of locally convex Now suppose a is an ordinal
topologies on E. number and (E,T ) a
's
has been defined for all ordinals S
Let
= lim (E"S)  i.e. T
S
defined by all 'S,S
is the projective limit topology a We define 'a as the topology which has
a neighbourhood base at zero consisting of all Ta closed convex balanced absorbing subsets of E.
By transfinite induction 'a
is defined for every ordinal a.
Since the cardinality of the
set of all convex balanced absorbing subsets of E is less than or equal to 21EI
it follows that there exists an ordinal
number e such that 'e = Tel for all
e > e. Since a locally l convex space is barrelled if and only if each closed convex
balanced absorbing set is a neighbourhood of zero it follows that
(E"e)
is a barrelled locally convex space.
Since ' t .::..
T
113
Holomorphic functions on balanced sets
the above construction shows that ' t ~ 's· As ' t is the weakest barrelled topology on E finer than, it follows that We now show by transfinite induction that (E"a) is a complete locally convex space for each ordinal number a. By hypothesis (E"l)
=
(E,,) is complete.
Now suppose (E"a)
is complete for all a< a l . Since the projective limit of complete locally convex spaces is complete the space (E, Tal)
lim
+~l
net in (E "
(E"a) is complete.
Hence ). al complete space (E,T ) al Now let V be (E,T )· al out loss of generality
Let (x ) be a Cauchy 13 13E: B
is also a Cauchy net in the 13 E: B and so converges to an element x in (x 13)
a 'a neighbourhood of zero in E. With 1 we may suppose that V is a Tal closed
Since (1:: ) is , al  Cauchy 13 13 E: B there exists 13 0 in B such that x13l  x13 E: V for all 131' 132 ~ 13 0 2 Since x132 x in (E, Tal) as 13 2   + '" and V is Tal closed it follows that xl3  x E: V for all 13 ~ 13 0 , Hence xl3  .... x in convex balanced subset of E.
(E"al) as 13> "'. This shows that (E"al) is complete.
,
By
transfinite induction (E"a) is complete for each ordinal number a and hence (E"s) = (E"t) is a complete locally convex space. This completes the proof. A modification of the above proof yields the following result and also gives an alternative description of the associated infrabarrelled topology. Proposition 3.6 let ' t and ' i
Let (E,,) be a locally convex space and be, respectively, the barrelled and infra
barrelled topologies associated with ,. then (E"i) is also complete.
If (E,,) is complete
If (E,T) is quasicomplete then
(E"t) and (E,T i ) are both quasicomplete. If (E,T) is sequentially complete then (E,T t ) and (E"i) are both sequentially complete.
We now look at various kinds of Schauder decompositions. These are generalizations of the concept of a basis and indeed a locally convex space has a Schauder basis if and only if it
Chapter 3
114
has a Schauder decomposition consisting of one dimensional subspaces. Definition 3.7 con~ex
A sequence of subspaces
space (E,T)
tEn}n of a locally
is a Schauder decomposition of (E,T) i f
each x in E can be written in a unique way as z:"" xn (i. e. n=l z: m x 11 )where Xn E: En aU n and i f the projections x = lim n=l m+ "" , E, Um (l:"" xn) = l:m Xn are aZZ continuous. Um : E n=l n=l
(Um);=l' form an equicontinuous family then we say that the decomposition is equiIf the sequence of mappings,
Schauder.
If E contains a fundamental family of seminorms,
~, such that p(l:"" xn) = n~:l p(xn) for every p in ~ and every n=l ~"" Xn in E then we say that the decomposition is absolute. The n=l decomposition is said to be shrinking ifC(En\i);;'=l is a
Schauder decomposition for E
B.
It is easily seen that every absolute Schauder decomposition is an equiSchauder decomposition.
We let4 denote the
set of all sequences of complex numbers (an)n such that lin lim sup lanl ~ 1. If {En}n is a Schauder decomposition n  ""
for (E,T) then we say {En}n is an JSchauder decomposition i f
z:"" Xn E: E and (an) n"" =1 E: .6 imply 1: anXn E: E. The decomposition n=l n=l is an ~absolute decomposition i f for every p E: cs(E) and (an)~=l €:,8 the seminorm p(~""
n=l
continuous.
Proposition 3.8
Xn) = ~"" n=l
\a \ p(x n ) is n
A Schauder decomposition {En}n of a locally
convex space (E,T) is equiSchauder i f and only i f there exists a fundamental system of seminorms (Pa)aE: A for
p (l ( x ) = sup P ~ un ( x) ) n N
(
for every a in A and x
Proof
(E,T)such that
(*) in E.
We first suppose that {En}n is an equiSchauder
decomposition of E and p is a continuous seminorm on (E,T)' Let q(x) as n
7
= sup p(un(x)) for every x in E. n
00
Since un(x)
+
x
and p is continuous it follows that q(x) is finite
115
Holomorphic functions on balanced sets
for every x in E. where m 5
sMP q(un(x))
for every x in E.
{XEE; q(x} ~ l } and
B
Then B
for every x in E
n we have
q(x) Let
Since u (u x)= u (x) m n m
{x EE; p(u
n{x n
o
t:
E;
Un 1
(x))
;(U
n
< 1
(x))
B'
~
= {x; p(x)
<
l}.
for all n} l}
(Il).
Sinc e (Un) n is an equicont inuous fami! y of mappings
nu n
n
(B)
1
is a neighbourhood of zero in E and hence q is a continuous seminorm on E.
Since un(x) + x as n +
in E this shows that q over a fundamental
~
p.
~
for every x
As p ranges over cs(E) q ranges
family of seminorms which satisfy (*).
This completes the proof in one direction. Conversely if E contains a fundamental norms,(p} CI
CI E
{x EE;
and hence
family of semi
A' which satisfy (*) then for every a in A.
Pa(x)
(un)~=l
~
I}
=
B
=n
u
1
(B
)
n n a is an equicontinuous family of mappings. Cl
This completes the proof. Proposition 3.9
A Schauder decomposition of a barreZZed
ZocaZZy convex space is an equiSchauder decomposition. Proof
Let {En}n be a Schauder decomposition for the
barrelled locally convex space (E,T)'
If P is a continuous
seminorm on E let q(x)
for every x in E.
= sHP p(un(x))
As in the previous proposition q is a seminorm on E.
n.
B
u
Let
n
= {x EE; q(x} < 1} =n{XEE; p(un (x)) < Since each n is a continuous linear mapping {x EE; p(un(x}} 2. I} is a
closed subset of E and hence B is a closed convex balanced absorbing subset of E.
Since E is barrelled B is a neighbour
hood of zero and hence q is a continuous seminorm on E.
As
in the proof of the preceding proposition it now follows that q ~ p and q(x} = sMP q(u (x)). n An application of proposition 3.8 now completes the proof. The above method of proof is also used in the following proposition.
Chapter 3
116
Proposition 3.10 If {En}n is an 4Schaudep decomposition fop the bappelled locally convex space (E,T) then {En}n is an Aabsolute decomposition.
Proof
Let p
E cs(E)
and let (an)~=l
E"s.
The sequence
2 (n 2a )'" also lies in 4 and so if 1:'" x £ E then 1:'" n a n x n n=l n n=l n n=l 2 also lies in E. is a bounded subset of E. Hence {n a, x } "'' n n n=l 2 '" 1 Therefore s}lP pen a x ) = M <'" and Z;oo p(x ) < M·fi=l n n n n=l I a n I n2 Z;oo finite. is We have just Let P(~:l x ) n=l I a n I P (x n ) . n shown that p'" is a seminorm on E. Since B = {x E E', '" p(x) .2 1}
'"
=
n
n
{l:oo p(x ) < I } is a closed convex n m=l xm E E; ffi=l I a m I m balanced absorbing subset of E and E is a barrelled space it
follows that B is a neighbourhood of zero and hence '" p is a continuous seminorm on E.
This completes the proof.
We now look at associated topologies for spaces with an Aabsolute decomposition. Proposition 3.11
If {E .T} n
(E,T) then {E n fop (E, Tb) .
fop
,T
}
b n
A
Let B = (x )AEf
Proof of (E.T).
n
is an S~bsolute deeomposition
is also an 5absoZute decomposition
denote an arbitrary bounded subset
For each A in r let xl. = fi:1 x~
in (E.T).
P is a T continuous seminorm on E and
'" (1:00 x) = Z;"'1 P 1'!=1 n n= norm on E. Hence
s~p ~:1
(an) n 2 1a I p(x ) is also a T n n
n21anl
p(x~)
bounded subset of (E,T).
=
M < '"
£
~
~ontinuous
2
A
and {n a x}
, is a
If q is a Tb continuous seminorm
< Ml ~=1 ",00
q(x n )
1 n
<
(E.T b ) as A
0
~
as A
> '"
<'"
00
Hence the 2 is bounded on the T
bounded subsets of E and so is Tb continuous.
~
semi
n n n./\
on E then the above shows that sup[n 2 la Iq(x")] = M 1 n,A n n
that x~
If
then
in (E, T ) if xl. b
"'.
To complete the proof we must show that L;n x + Z;'" x in (E,T ) as n +00. b m=l m m=l m If p is a T continuous seminorm on E and x
This also shows 0 in
r~
117
Ho[omorphic [unctions on balanced sets
2 EOO EOO 2 Eoo 2 ) m=n Xm) ~ s~p m=n n p(X m) ~ m=l m p(X m < + 00. 2 oo Hence {n E x}oo 1 is a bounded subset of (E,T) and of m=n m n= (E'Tb)' If q is a Tb continuous seminorm on E then
pen
q(x as n
~~ll +
xm) = q( ""
~:n
x m) =
~2
q(n
2
~:n
xm)
+ 0
This completes the proof.
The situation for the associated barrelled topology is slightly more complicated.
This is due to the fact that we
do not, in general, know if a Schauder decomposition, {En} , n for a locally convex space, (E,T), is also a Schauder decomposition for the associated barrelled space (F,T ). t are t topologies on E i.e. if (E,T)' = (E,T )' and this t and only if (E,T)' is quasicomplete for the weak a positive answer to this problem if T and T
aCE' ,E).
We obtain compatible occurs if topology
On the other hand we do have the following unique
ness result; if {En}n is a Schauder decomposition for (E,T) and if each En is a barrelled space then there exists at most one barrelled topology on E which coincides with T on each En and which has {En}
n
as a Schauder decomposition.
Although we do not know if such a topology exists the above says that when it does it must be the associated barrelled topology.
We prove a less general result which is
adequate for our applications. Proposition 3.12
Let {En}n be anJ'decomposition for
(E,T)
and suppose each En is a barrelled locally conVex space.
If
there exists ~ barrelled topology TI on E which coincides with
T on each En and i f {En}n is a Schauder decomposition for (E,T l ) then TI = Tt and a fundamental system of seminorms for (E,T t ) is given by alZ seminorms, p, which satisfy the following conditions; (a)
plEn is Tcontinuous peE"" x ) = E"" p(X) for every n=l n n=l n E"" x in E. n=l n Proof By hypothesis T ~ T ~ TI and hence t (b)
118
Chapter 3 {E
}
n n
is a Schauder decompos
it follows that it is also a Schauder decomposition for (E,T t ). Since {En}n is an ,Jdecomposition for CE,T) it is also an .Jdecomposition for (E,T t ) and (E,T l ) and hence by proposition 3.11 it is an Jabsolute and in particular an absolute decomposition for both of these topologies.
Hence both of these topologies are
generated by seminorms which satisfy (a) and (b).
On the
other hand if p is a seminorm on E which satisfies (a) and (b) then V
=
{x E E; p(x) ::.. l } is a Tclosed convex balanced
absorbing subset of E and hence p is a T
continuous semit norm. Hence Tl = Tt and Tt is the locally convex topology on E generated by all the seminorms on E which satisfy (a) and (b) .
We now look at decompositions of the strong dual. If {E} is an ~absoZute decomposition for n n CE,T) then {(E )13'}n is anJabsoZute decomposition for (E,T)13" n
Proposition 3.13
A
Proof cr
=
Let B m (x )AEr be a bounded subset of E and let
(cr)
n n
E J(we may and shall assume without loss of
Since {E} is an g enerali ty that cr n > 1 for all n). n n Jabsolute decomposition it follows that 2 B' = (n cr x A) and B" = {n 2 t;'" x A) are bounded subsets n n n,A m=n m n,A of E. n
If TEE' we let T B" is bounded Itr 
M~ll
Tmll B =
for every positive integer n.
= TIE
Since
n
s~p
IT(M:n
x~)1
::..
~2I1TIIB"
+
0
as n
> '"
If Te E E' for each a in some directed set Q and Bn is the projection of B in En then liTanllBn
= S~pITa(X~)
Hence, since En T~
,
IITaIlB' for all n. I < l2 n cr n is a complemented subspace of E,
in (E n )13' as a ' for each n ifTa , o i n E '. 13 This shows that {(E )13'}n is a Schauder decomposition for E '. n 13 To complete the proof it suffices to notice that 0
00
H%morphic functions on balanced sets
119
1
i12 for every T in E'. Corollary 3.14
An/iabsolute decomposition of a locally
convex space is a shrinking decomposition. Further properties of associated topologies and Schauder decompositions are given without proof as the need arises. For further details we refer to the Notes and Remarks at the end of this chapter. Apart from the immediate applications of the above results which we give in this chapter we will also find that the techniques developed to obtain these results are useful in studying holomorphic functions on certain nuclear sequence spaces. §3.2
EQUISCHAUDER DECOMPOSITIONS OF (H(U;F) ,T). In this section U will denote a balanced open subset of
a locally convex space E and F will denote a Banach space (Most of the results, however, can be extended to arbitrary quasicomplete locally convex spaces). Proposi~ion
If U is a balanced open subset of a locally
3.15
convex space E, F is a Banach space, f EH(U;F) and a
=
(an)n EJ then g
l:'"
a
n=o
H(U;F)
E
n
Let K c:. U be compact.
We may suppose without loss
of generality that K is balanced, then there exist A>l and V, a convex balanced neighbourhood of zero, such that A(K+V) C U and
Ilf II A( K+ V)
M
< '" •
By the Cau c h y in e qua l i tie s
Ildnf~O)
lI:qK+V) .:::. M for all n. n. Since (an) E ~ there exists C > n
0
such that
lanl .:::. C f;A)n for all n. Hence II gil K+V
.:::.
;="'0 ani 1
<
Chapter 3
120
'" I+A n (  ) <"'. n=o 21.
< C.M Z
Thus g is a locally bounded function on U and since it is Gholomorphic it
is
in H(U;F).
This completes the proof.
If U is a balanced open subset of a locally
Proposition 3.16
conVex space E,F is a Banach space and fEH(U;F) then the Taylor series expansion of f at the origin converges to f in
Proof
For each positive integer m let
An V
{x
m
E:
. U)
n
2nd \1
£(0)
n.1
the interior of V . m
Since
(Wm)~=l
ition shows that
ex)
~ ~ m for all n}
2
00
(n ) n=l E
8
and let W denote m
the preceding propos
is an increasing countable open cover
of U.
Now let p be a To
exist
a positive integer mo and C>o such that
for every f P <
(f
_
~jf(O))=
zn j=o
C
II
in H (U; F) .
continuous seminorm on H(U;F).
Ilfllwmo
Hence for all n we have
ZOO
j!
p(f)~C
There
P(j=n+l
dj~~o)) J.
cijf(o)
Z'"
j=n+l
~ C Z'" j =n+l
j !
j2
Therefore p(f 
",.n ~ ]=0
cijf(O))  + 0 as n j !
~
'"
and this completes the proof. Theorem 3.17
Let U be a balanced open subset of a locally
convex space E and let F be a Banach space. {p(n E ; F),
T
} '" 0
w n=
decomposition for Proof
Then
is an.8 decomposition and an,J absolute (H(U;F) ,TO)
By proposition 2.41
(6'(n E ;F),T w )
is a closed
complemented subspace of (H(U;F),T O ) and so proposition
121
Holomorphic functions on balanced sets 3.16 implies that for
(H(U;F) ,TO)'
{@(nE;F),T}~
is a Schauder decomposition w n=o Proposition 3.15 implies that it is an
Jdecomposition and since To is a barrelled topology proposition 3.10 implies that it is an Jabso1ute decomposition.
This completes the proof.
We now obtain the same result for the TO and Tw topologies and their associated borno10gica1 topologies.
Let U be a balanced open subset of a
Proposition 3.18
locally convex space E, let F be a Banach compact subset of U and let (a) p (f)
is
T
o Proof
~:o I an

I
dnf(o) t' It n! IK
€
g.
space~
let K be a
Then the seminorm
n n
continuous on H(U). We may assume without loss of.genera1ity that K
is a balanced subset of U.
Choose A>l such that AK is a
compact subset of U and no a positive integer such that lanl
(l;A)n for all n
<
For any f I an
in H(U;F) we have by the Cauchy inequalities
I Ildn~fO) ~K ~
Hence p (f)
no 1 <
[l:
n=o
for every f
l:~
nno
(1+A)n 1 2 An
IlfIIAK
Ia n I
in H(U;F)
and this completes the proof.
Let U be a balanced open subset of a locally
Theorem 3.19
convex space E and let F be a Banach space.
{6'
(n E ; F),
T
}
00
o n=o
Then
is an.¢ decomposition and an'& absolute
decomposition for Proof
(H(U;F),T )' O By proposition 3.16,
since TO
~
TO'
the Taylor
series expansion at the origin of a ho1omorphic function converges to the function in the compact open topology.
(~(nE;F)'T ) is a closed complemented subspace of o (H(U;F),T) (proposition 2.40) this shows that {6>(n E ;F),T}"" o 0 n=o is a Schauder decomposition for (H(U;F) ,TO)' Proposition
Since
3.15 implies that it is anJdecomposition and proposition 3.18 shows that it is an.,8abso1ute decomposition.
This
122
Chapter 3
completes the proof. Corollary
Let U be a balanced open subset of a locally
3.20
Then {~(nE;F)}~=o
convex space E and let F be a Banach space.
is anS decomposition and an.,& absolute decomposition for (H(U;F),To,b) i f each ~(nE;F) is given the bornological
topology associated with the compact open topology. Proof
Apply proposition 3.11 and theorem 3.18.
Let U be a balanced open subset of a
Proposition 3.21
locally convex space E,let F be a Banach space,
let p
continuous seminorm on H(U;F) and let (a) E J. w . n n Then the seminorm 'n (d f{02) is T continuous on H(U;F) . L: oo P (f) I a I p n! n w n=o Proof Suppose p is ported by the compact bal anced subset K
be a T
'"
We show that p'" is also ported by the same compact set.
of U.
Let V be a neighbourhood of K which lies in U.
Choose A> 1
and a balanced neighbourhood of zero W such that K c. A(K+W) C V. Choose a positive integer no such that n
~
no'
la n I
(l+A)n for all 2 such that
<
There exists a positive number C(W)
p (f) 2. C (W)
Ilf IIK+W
for every f
Hence,
in H (U; F) .
for every
f in H (U; F), we have
2. C(W)
p(f)
<
L: OO
n=o
Ia n I Ia n I
C (W)
Ia n I and
p
is a T
Theorem 3.22
w
continuous seminorm on H(U;F).
Let U be a balanced open subset of a locally
convex space E and let F be a Banach space.
Then
{6'(n E ; F) ,T } 00 is an J decomposition and an"g absolute w n=o decomposition for (H(U;F),T W) '
Proof
By proposition 3.16,
since To
~
T
W
'
the Taylor
series expansion at the origin of a ho1omorphic function converges to the function in the T
topology. By proposition 00 w 2.41 {\p(nE;F),Tw}n=o is a Schauder decomposition for (H(U;F),TW ) It is an/~'decomposition by proposition 3.15 and proposition
3.21 shows that it is an,Jabso1ute decomposition.
This
Holomorphic functions on balanced sets
123
completes the proof. Corollary 3.23 {~(nE;F),,}oo isanJdecomposition and an w n=o JabsoZute decomposition for (H(U;F)" b). w, Proof
Sin c e ,
I
<, b <, an d , w w, 0 w ~(nE;F)
 , 0 I~(nE;F)
for all n it follows that,
~(nE;F)
w, b induces the, w topology on The required result now follows by apply
for all n.
ing proposition 3.11 and theorem 3.22. The existence of anJabsolute decomposition of H(U;F) by spaces of homogeneous polynomials is adequate for all our applications in this chapter.
However we shall need,
in
chapter 4, a slightly stronger result which appears to hold only for the compact open and the
'w
topologies.
Let U be a baZanced open subset of a
Proposition 3.24
ZocaZly conVex space E, let F be a Banach space and let p be a , continuous seminorm on H(U;F), ,= '0 or
Two
Then
there exists A>l such that n
n
'V
p
(Loo d f(o)) n=o n!
An p(d f~o)) = Loo n=o n.
is also a , continuous seminorm Proof
I f P is a TO continuous
on H(U;F). seminorm on H(U;F)
and
P (f) ~ C IIf 11K for every f in H(U;F) then we may choose A>l For every f =L oo d f(o)cH(U·F) such that AK is a subset of U. n=o n! ' 00 oo 'V cinf(o) ) L An (dnf(o»)<);oo CAn Ildnf(O) II p (~=o n=o p n!  n=o n! K n! ~n
~
~n
ii:o
C
'V
and hence p is
'0
lid
~fo)
IIAK
continuous.
This completes the proof for
the compact open topology. Now suppose p is a TW continuous seminorm on H(U;F) is ported by the compact balanced subset K of U. such that AK is a compact subset of U. balanced subset of U which contains AK.
a
II£fi
Choose A>l
Let V denote an open There exist a>l and
W a balanced neighbourhood of K such that AK Let C (W) >0 be such that p (f) ~ C (W) oo H(U.F). If L f(o) c H(U·F) , n=o n! '
which
w
c:
aAWC:VCU.
for every f in
124
Chapter 3
00
L:
n=o C (W)
~:o ~n Ilfil aAw Ilf Ilv .
<
C(W)
<
C .
Hence p is ported by AK and so it is T
w
continuous.
This
completes the proof. We complete this section by considering spaces of germs. Theorem 3.25
If K is a balanced compact subset of a locally
conVex space E and F is a Banach space then {~(nE;F) ,T }oo
W n=o
is an g decomposition and an
t8 absolute decomposition for
H(K;F). Proof
If f EH(K;F)
subset U of E.
then f EH(U;F)
for some balanced open
Since H(K;F)=lim, U (H(U;F) ,TW)
as U ranges
over all balanced open subsets of E containing K and since the Taylor series converges in(H(U;F),T ) (theorem 3.22) it w follows that the Taylor series of f at the origin converges to fin H(K;F).
By proposition 2.58
(6)(n E ;F),T) is a closed w complemented subspace of H(K;F) and hence {~(nE;F) ,T}oo is w n=o a Schauder decomposition of H(K;F). Proposition 3.15 shows that it is an,8 decomposition and an application of proposition 3.10 completes the proof since H(K;F)
is a
barrelled locally convex space.
§3.3 APPLICATIONS OF GENERALISED DECOMPOSITIONS TO
THE STUDY OF HOLOMORPHIC FUNCTIONS ON BALANCED OPEN SETS The results of the two previous sections are applied to H(U;F) where U is a balanced open subset of a locally convex space and F is a Banach space.
The first part of this section
is devoted to topologies associated with the TW topology.
Our
first result motivated the introduction of associated topologies in the theory of holomorphic functions on locally convex spaces.
125
Holomorphic functions on balanced sets Theorem 3.26
Let U be a balanced open subset of a locally
convex space E and let F be a Banach space.
= Proof
Since (H (U; F) ,T ..)
T
On H(U;F)
w, ub.
is an ul trabornological space it
suffices to show that To is the barrelled topology associated By proposition 2.41 TW and To agree on6>(n E ;F)
with TW' each n.
for
An application of proposition 3.12 now completes the
since {~(nE;F),T}oo is anJdecomposition for both w n=o (H(U;F},To)by theorems 3.17 and 3.22.
Proof
(H(U;F},T ) and W
Proposition 3.12 also shows that To is the finest topology for which we have absolute convergence of the Taylor series expansion and which coincides with T homogeneous polynomials.
on spaces of w Formally this is expressed as
follows.
Let U be a balanced open subset of a
Proposition 3.27
locally convex space E and let F be a Banach space.
The To
topology on H(U;F) is genepated by all seminopms, p, which satisfy the following conditions; An o (a) p(f) = li:o p (d )) fop evepy f in H(U;F)
!f
(b)
PI
is
6'(n E ;F)
T
w
continuous.
The following lemma is an immediate consequence of the existence of an;5 absolute decomposition.
An analogous
result for the compact open topology is also true. Lemma 3.28
convex space
Let U be a balanced open subset of a locally ~
and let F be a Banach space.
Let
(fa)a
E
r be
a TW (resp?ctiveZy Tw,b,T ) bounded net in H(U;F). Then o f + 0 as a    r 00 in (H (U; F) ,T ) (respective ly a w (H(U;F),T b)' (H(U;F),T o )) i f and only i f An w, d f (0)/ + 0 as a     7 00 in (~(nE;F) ,T )for evepy nona , w n. negative integer n. This means,
in particular, that T ,T W
w,
b and
T~
induce the
I.)
same topology on the TO bounded subsets of H(U;F).
Theorem
3.26 implies, among other things, that Tw and To define the same convex balanced complete bounded subsets of H(U;F).
Chapter 3
126
Using lemma 3.28 we show that the same result holds for compact balanced convex sets.
Let U be a balanced open subset of a
Proposition 3.29
locally convex space E and let F be a Banach space.
Then the
convex balanced compact subsets of (H(U;F)"w) and (H(U;F),,~)
with,
coincide and '0 is the Kelley topology associated
on H(U;F).
w
Since '0
Proof
~
'w it suffices to show that any convex
By K of (H(U;F)"w) is '0 compact. theorem 3.26 K is a complete balanced '0 bounded subset of balanced compact subset
I f (f(l)(lE:r is a net in K then it contains
H(U;F).
convergent subnet.
a ,
Hence K is a '0 compact subset of H(U;F).
convergent.
w
By lemma 3.28 this subnet is also '0 Since
'0 is an ultrabornological topology it is also a Kelley topology and hence
,~
\,)
=
T
K
w,.
One can also show that 'w,b is the infrabarrelled topology associated with Tw on H(U;F). are,
Thus we see that there
in general, two types of topologies that we may associate
with TW'
On the one hand there are the associated barrelled,
ultrabornological, barrelled and bornological, and Kelley topologies all of which are equal to TO and the associated infrabarrelled and bornological topologies which are equal to Tw,b'
It is an open question whether or not these two
topologies coincide i.e.
is 'w,b = TO?'
Theorem 3.26 and
proposition 3.29 indicate that they are very close to one another.
The following result gives necessary and sufficient
conditions under which these topologies coincide and we shall in this and later chapters, encounter various sufficient conditions for their equality. Proposition 3.30
Let U be a balanced open subset of a
locaZly convex space and let F be a Banach space.
The
following are equivalent on H(U;F); (a)
'w,b
(b)
(c)
'0 and w define the same bounded sets TO and TW define the same compact sets~
Cd)
'wand T& induce the same topology on Tw bounded sets
,
127
Holomorphic functions on balanced sets (e) ( f)
Tw,b is a barrelled topology Tw,b is the finest locally convex topology for which the
Taylor series expansion at the origin converges absolutely and which induces the
T
w
topology on ~(nE;F) for every
positive integer n, (g)
E ((f'(~E;F),T )' for every nonnegative integer n n"" 'dnf(o) w L:"" anf(o) and ~=o Tn C n ! ) converges for every f n=o nl in H(U;F) then ~:o Tn E (H(U;F),TW,b)'·
if T
Proof
(a), (b), (e) and (f)
and proposition 3.27.
are equivalent by theorem 3.26
(a)=9(c) by lemma 3.28.
I f (c)
holds
and B is a TW bounded subset of H(U;F) which is not To bounded then there exists a To
continuous seminorm p and
(fn)n' a sequence in B, such that p(f n )
   + ""
as n
  + "".
{~/Ip(fn)}""_ ufo} is TW compact but not TO bounded. This contradiction ~h&ws that (c) ::::::}(b). Cc) and (d) are
The set
equivalent by lemma 3.28.
Now suppose (a) holds and the
sequence {Tn} satisfies the conditions of (g)~ By proposition 3.15):;"" ITn rdnf(o) ) I < '" for every f ):;'" dnf(o) E H(U'F) n=o nl n=o n! ' By proposition 3.27 p(f) = L:"" IT (anf(O)) I n=o n n! defines a TO and hence a Tw,b continuous seminorm on H (U; F) • Since
I'n 00
I ~=o
I~
Tn
):;"" IT C d f(o)) n=o n nl
it follows that):;'" Tn E (H(U;F),T b)' n=o w, Conversely if (g) is satisfied then (H(U;F);T w, b)'
= (H(U;F)
I = p(f)
and hence (a)
=?
(g)
,T~)'
and since T w, b is a Mackey topology (it is infrabarrelled) this implies that Tw,b = To and (g) ===} (a).
u
This completes the proof.
Some analogues of the above results can also be proved for the compact open topology.
The results, however, are not as
complete in this case since ((p(nE;F),T ) is not in general a o barrelled locally convex space. We give one example. Proposition 3.31
Let U be a balanced open subset of a
locally convex space E and let F be a Banach space. following are equivalent on H(U;F);
The
128
( a) (b)
Chapter 3
T
0,
(i)
b is a barrelled topology~
(~(nE;F),T
(ii) i f Tn
0,
b) is barrelled for each integer n,
w(nE;F) ,T b)' for each nand , 0, dnf( ) 00 anf(o) Z:_o Tn ( 0 ) < 00 for each f = Z n n! n=o n! E
in H(U;F) then ZOO T n=o n (c)
(i) (ii )
(~(nE;F),T T
0,
0,
b is the
E
(H(U;F) ,T
0,
b)',
b) is barrelled for each integer n,
finest locally convex topology on
for which the Taylor series converges and
H(U;F)
which induces on
@(nE;F) the To,b topology for each n.
We now introduce a weak form of completeness  Taylor series completeness  which allows us to extend various topological properties of spaces of homogeneous polynomials to holomorphic functions on balanced open sets.
Let E and F be locally conVex spaces and
Definition 3.32
let U be a balanced open subset of E. space (H(U;F),T) is T.S.
T complete
The locally convex
(T.S.
~
Taylor series)
i f the following condition is satisfied; i f (P n )"" is a n=o sequence of homogeneous polynomials~ P E ~ (nE;F), and n
00
Z p(P) < 00 for each n=o n );00 P sH(U;F). n=o n
continuous seminorm p,then
T
We have already seen examples of T.S. completeness. For example theorem 2.28 says that H(U;F) is T.S. Tp complete if U is a balanced open subset of a Banach space and T is p the topology of pointwise convergence. Let TI and T2 denote two locally convex topologies on H(U;F) and suppose TI is also T.S.
~
T2 ·
If H(U;F)
Tl complete.
is T.S.
T2 complete then it
The following result describes a
situation in which the converse holds. Lemma 3.33
Let U be a balanced open subset of a locally
convex space and let F be a Banach space.
If
T
is a locally
convex topology on H(U;F) and {(?(nE;F),T}~=o is an,.8absolute decomposition for H(U;F) then H(U;F)
is T.S. T complete i f
and only i f H(U;F) is T.S. Tb complete Proof
Suppose H(U;F)
is T.S.
Tb complete.
Let
(Pn)~=o
129
Holomorphic functions on balanced sets be a sequence of homogeneous polynomials, P £ ~(nEiF), and n
suppose ~:o p(P ) < 00 for every T continuous seminorm p on n H(UiF). The sequence {p} is a T and hence a Tb bounded n n
subset of H(UjF).
00
Since {(f'(nEjF),Tb}n=o is also anJabsolute
(H(U;F),T ) (proposition 3.11) we have b for every Tb continuous seminorm on H(U;F).
decomposition for <
k:o p(P n ) oo Hence E Pn£H(U;F) and H(U;F) n=o completes the proof. 00
is T.S.
')" complete.
Let U be a balanced open subset of a locally
Corollary 3.34
convex space and let F be a Banach space. T.S.
T
T.S.
T
o 0,
This
(respectively
T
Then H(U;F) is
complete i f and only i f it is
)
w
b (respectively
Proposition 3.35
T b) complete. w, Let U be a balanced open subset of a
locally conVex space and let F be a Banach space.
sequence
(~n)n
Hence k:o q
in
n!
r.
(dnf~n(o))
<
n!
on H(UiF)
If H(U;F)
00
and any sequence An that ~:o s~p q (d f~(o)) <
for every To continuous seminorm q (~n)n 00
in
r.
It is now easy to see
for any To continuous seminorm
n'
q and this completes the proof. Our next result shows the connection between T.S. completeness and completeness. Proposition 3.36
convex space
E
Let U be a balanced
and let
F
open subset of a locally
be a Banach space.
Let')" be a locally
convex topology on H (U; F) such that {i?(n E ; F) ,Tl is an,J absolute (e.g.')"=')" ,T b'')" ,T b or T~). o 0, W w, u (H(U;F),T) is complete (respectively quasicomplete~
decomposition for H(U;F)
Then
sequentially complete) i f and only i f (~(nE;F) ,T) is complete (respectively quasicomplete, sequentially complete) for every nand H(U;F) is T.S. T complete.
Chapter 3
130 Proof
The conditions are obviously necessary.
are sufficient.
cases are handled in a similar fashion. Cauchy net in
a
>
H(U;F). f
=
We may suppose p(f)
);00
=
~:o p(an~~o))
Given E > E H(U;F). An dnf (0) d f 13 (o)) < p( a
n!
that );00 n=o
is a Cauchy net in a Er E f(nE;F)
as
Let p be a T continuous seminorm on
anf(o)
n=o
(fa)aEr be a
for each n and hence anf (0) + P a n n!
for each n.
00
Let
(H(U;F),T). Then {anfa(O)} n!
(p(nE;F),T)
We prove they
We consider only the complete case, the other
0
for every
we can find a
o
E r such
n!
n!
for all a,13 Er, a> k
Hence ~=o p(
dnf a (0)
a
n! positive integer k.
k
00
anf
(0)
) + E for all k and so n! for every T continuous seminorm p. Since
); p(P) < ); n=o n  n=o p ( <
and every
In particular
00
~:o p(P )
o
no
n is T.S.T complete this implies that f =~:o P n E H(U;F). An d f (0) The above also shows that );"" P( II ~ E for all n=o n! a > a and hence f + f as a + 00. This completes o a the proof. H(U;F)
Our aim now is to show that
(H(U)'T
W
)
is complete when
ever U is a balanced open subset of a metrizable locally convex space.
Since (H(U),T
O
)
is complete for any open subset
U of a metrizable locally convex space proposition 3.36 implies that H(U)
is T.S.T
balanced open set U. show that
and hence T.S.T complete for any o w Hence to prove this result we must
(lP(n E) ,T ) is complete for any positive integer n. w
First we need some preliminary results which are also of independent interest. Proposition 3.37
convex kspace.
Let U be an open subset of a locally Then
(H(U),T O ) is a semiMontel spaae
the Tobounded subsets of H(U) are relatively compact).
(i.e.
131
Holomorphic functions on balanced se ts Let t(U)
Proof
denote the continuous complex valued
functions on U endowed with the compact open topology. B be a subset of H(U).
Now (H(U), TO)
Let
is a closed subspace of
k(U) and hence B is a closed bounded (respectively compact) subset of (H(U),T ) if and only if it is a closed bounded O (respectively compact) subset of ),(U). By using Taylor series expansions we see that any TO bounded subset of HCU)
is equi
continuous on the compact subsets of U and hence an application of
Ascoli~
theorem completes the proof.
If U is an open subset of a J:;1hz space then
Corollary 3.38
(H(U),T o ) is a FrechetMontel space. A 3HlYl space is a kspace and hence CH(U) ,TO)
Proof
Montel space.
Example 2.47 shows that
(H(U),T O )
is a semi
is a Frechet
space and this completes the proof. Corollary 3.39
If U is an open subset of a metrizable
locally convex space then
(HCU),T ) is a semiMontel space. O
The above results and similar Montel type theorems could also be proved by using Schauder decompositions.
Some of
these are to be found in the exercises at the end of this chapter. We now need a linear result which will also be useful in chapter 6. Proposition 3.40 Let T , T2 and T3 be three Hausdorff I locally convex topologies on a vector space E such that (a)
TI~T2~T3;
Cb)
(E,T I ) is a bornological DF space (or equivalently a
countable inductive limit of normed linear spaces) with a countable fundamental system of closed convex balanced bounded sets (c)
Cd)
(B )
n n
;
CE,T ) is a barrelled locally convex space; 2 Bn is T3 compact for all n.
Then TI Proof
= T2 · A fundamental system of neighbourhoods of zero in
CE,T ) is given by sets of the form I
roo
n=l
A B
n n
= {Em n=l
Ax . n n'
X
n
£
Band m arbitrary} where A n n
Chapter 3
132
is positive for all n EN. Let V = ~:l AnBn' An> ~ denote the algebraic closure of V in E, i.e. 'V
{x E E; AX E V for
V
2. A
0
<
and let
0
Since Bn is a compact sub
I}.
set of (E"3) it follows that ~~l AnBn is also a compact subset of (E"3) and hence a closed subset of (E"2) for every positive integer k.
¢
Now let x
'V
V.
Then there exists A> 1 such that x ¢ AV and hence x ¢ AL: k A B for every n=l n n integer k. For each k choose ¢'kdE"2) I such that
hence a relatively weakly compact subset of (E"2)
I.
If
is a limit of a weakly convergent subnet of the sequence {
I
(x) = A and
V'2 c:. (1+ E) V for every E
1. >
Hence \i'2 = ~ and so
o.
Since V'2 is convex balanced
absorbing and '2 closed it is a neighbourhood of zero in (E"2) and so every '1 neighbourhood of zero contains a '2neighbourhood. This shows that '1 = '2 and completes the proof. Proposition 3.41 Let E be a met!'izable locally convex space and let n be a positive integer.
On i?(n E)"
W
=,
0,
t
(i.e.
,
W
is the barrelled topology associated with, ). o
Proof
((p(n E) , , ) is a bornological DF space with w
fundamental system of bounded sets
lip IIv m 2. I} where Vm ranges over a fundamental neighbourhood system of zero in E conSisting of
Bm = {P
£
(p
(n E);
closed convex balanced sets. By Corollary 3.38 Bm is a compact subset of (~(nE)" o ). Since, w is a barrelled topology 'w ~ 'o,t ~ in proposition 3.40. ,
Ul
=,
0,
t and this completes the proof.
Corollary 3.42 then
Let 'w = '1' 'o,t = '2 and '3 = ' 0 Since all the requirements are satisfied
'0'
If E is a metrizable locally convex space
(~(nE),'w) is a complete locally convex space for every
non negative integer n.
Proof

Since (fCnE)"
0
) is complete proposition 3.5 implies
that PenE) endowed with the associated barrelled topology is also complete.
The associated barrelled topology is 'w by
proposition 3.41 and this completes the proof.
133
Holomorphic functions on balanced se ts As a further corollary we obtain a generalization to homogeneous polynomials of the well known linear characterization of distinguished metrizable spaces.
If E is a metrizable locally convex space then
Corollary 3.43
the following are equivalent; (~(nE),'o)
(a)
(respectively
(~(nE),S)) is a barrelled locally
conVex space,
(6'(n E )"
(b)
o
)
(respectively
CCr(n E) ,S)) is a bornological
locaZly convex space, ((p(n E)" ) (respectively ((p(nE) ,S)) is an ultrao bornological locally convex space.
(c)
Proof
It suffices to notice that
on (p(nE) 'w
=
Sb
and, since,
=
St
=
o
.::. S
,
,
=,
uJ o,b o,t we also have
Sub'
Corollary 3.44
If U is a balanced open subset of a metriz
able locallb convex space then (H(U)"w) and (H(U)"8) are both complete locally convex spaces. Proof
It suffices to apply proposition 3.36 and corollary
3.42. If U is a balanced open set and (H(U) "w) then
is complete
(H(U)"6) is also complete, we may prove this in three
different ways; (a)
since '8 is the barrelled topology associated with Tw (theorem 3.26) we may apply proposition 3.5 which says that the barrelled topology associated with a complete locally COnvex topology is also complete,
(b)
if (H(U)',w)
is complete then H(U)
is T.S.
(proposition 3.36) and hence since, '8 1'.S. '6 complete.
~
'w complete
'w' H(U) is
Since '0 and 'w agree on(fl(n E)
for all
n we may then apply proposition 3.36 to complete the proof, (c)
if (H(U)"w)
is complete then H(U)
is T.S.,w complete
and hence also T,S"wwb complete (corollary 3.34)
and an
application of proposition 3.36 completes the proof. If U is an open subset of a quasinormable metrizable
Chapter 3
134
(H(U),.w) is complete.
space then it is known that
However
the general problem for open sets in metrizable spaces is still open.
We return to this question in chapter 6. Results similar to the above may also be proved for
holomorphic germs by using the same techniques.
In this
manner we obtain the following results. Proposition 3.45
1et K be a compact baLanced subset of a
LocaLLy convex space E and Let F be a Banach space.
Then
H(K;F) is a complete (respectively quasicompLete,sequentiaLly comrlete) Locally convex space i f and onLy i f (f(nE;F)"w i is complete (respectively quasicomplete, sequentiaLly compLete) for aLL n and for any sequence of homogeneous polynomials 00 .on (P) ,P E (f( E;F) ,J: p(P)< 00 for each continuous semi· n n=o n n=o n 00
norm p on H(K;F)impZies
P
J:eo
n=o
n
E
H(K;F).
If K is a compact baLanced subset of a
Corollary 3.46
metriBable localLy conVex space E then H(K) is a complete locaLLy conVex space. By corollary 3.42
Proof
(!p(n E )
I' )
is complete for all n. w If (Pn)n=o is a sequence of homogeneous polynomials, 00
E lP (n E), and n=o J:"" p(P) < eo for each continuous seminorm n on H(K) then {P}"" is a bounded sequ~nce in H(K). Since P n n=o H(K) is a regular inductive limit (proposition 2.55) there
P
n
exists a neighbourhood V of K and sliP
I Pn II AV
~:o
P
~l
"
EH(K).
n the proof.
~
>
1 such that
1 and so ~ < 00 Hen c e ~ : 0 " Pn "V ~ M ~ : 0 An application of proposition 3.36 now completes < "".
In the above corollary we used the regularity of H(K) which was proved for arbitrary compact subsets of a metrizable This result can also be proved independ
space in chapter 2.
ently for balanced compact sets by using Schauder decompositions; specifically oneuses the seminorms
An
p(f)
"J:'"
n"o
I~f(ol n!
(x)1 n
where (xn)n ranges over all sequences which tend to K.
135
Holomorphic functions on balanced sets
In chapter 5, which deals with holomorphic functions on nuclear spaces, we shall see that regularity and completeness of spaces of germs are equivalent in a number of nontrivial situations.
We give here an example of a space of
germs which is not regular.
Later results will show that it
is also not complete. Example 3.47
~(N).
Let E
~n
we identify
For each positive integer n
n coordinates.
Let 0 denote the origin in
be a continuous seminorm on H(O). integer n such that if f
H(O)
£
u:
neighbourhood U of zero in
PI
space)
n
and let p
[(N)
We claim there exists an
and flu then p(f)
"
~n
o for some
IL
We may suppose
= o.
An
r:
p(d f~o))for every f in H(O). no n. is 'wand hence ' 0 continuous (E is a'j) 'j"1
by theorem 3.25 that p(f) Since
[(N) spanned by the first
with the subspace of
(f> (kE)
=
we can find for every integer k another positive
integer k'
such that if P
r:?
£
(kE)
~k'
and P I
"
0
then p (P) = o.
Hence if our claim is not satisfied then we can find a sequence of homogeneous polynomials, PJ'I is a of
. "
0
~J
'0
F o.
and p(P.) J
(Pj)~"l' such that
.• ) Jr.
bounded subset of H(
~
(N))
sinc~
u: {N)is contained and compact in
integer n,
Hence,
since
J
Let Q. = =.L J p(P.)
'0
for all
j.
en {Q.} J j=l
every compact subset
(n for some positive
'won H( u:(N))
(example 2.47).,
{Qj}~=l is a bounded subset of H(O). But p(Q.) J claim.
= j
j
for all
and this contradiction proves our x
For each integer n let fn((xm)m)
n
Since f
"lnx 1
H(0l[ nl)
for all n the above shows that {f}
subset of
H(O)~
n
n
=
0
in
is a bounded
If H(O) was a regular inductive limit then
there would exist a neighbourhood W of zero in which each fn was defined and bounded. 1
0 .,0, ) t W for all n. n' we conclude that H(O (N)) (
co
lim l (H V;J 0 V open
n
[(N) on
By our construction
Since this is impossible
II:
(V) ,"
IIv)
is not a regular inductive limit.
Chapter 3
136
We now consider linear functionals on HN(E) (for arbitrary E) and on H(E) for E a metrizable locally convex space with the approximation property.
We then combine these
results to show that T = T on H(E) when E is a Frechet o w nuclear space and (H(E) ,TO)'S ;;; H(OE'S) also under the same conditions.
These results generalise results already proved
for homogeneous polynomials in chapter 1 (proposition 1.61). The proofs use Schauder decompositions and estimates previously obtained in proving the corresponding result for homogeneous polynomials.
The results presented here are relatively recent
as these topics are currently the object of research.
We
discuss here two different situations and it is probable that a general theory which covers both situations simultaneously will appear in the not too distant future.
A similar theory
for balanced open sets has not yet been developed. Definition 3.48
Let E be a locally convex space.
If Y is a
convex balanced open subset of E we define An to n HN(Y) = {f E H(Y); d f(o)€u N( E) for each nand 00
'ITy (
f)
l:
00
= n=o
'IT ( Y
an f ~ 0)) n.
< oo}.
(H~(Y),'ITy) is a Banach space. HN(E), t~e nuclearly entire . . A"\ " ( n E) ad funct1ons on E, is def1ned as {f E H(E); d n f(o)wn N and for each compact subset K of E there exists an open subset
Y of E, Key, such that 'ITy(f)
<
oo}.
The 'ITo topology on HN(E) is generated by all seminorms An An oo 'ITKCf) = 'ITKCE d f(o)) _ E'" 'IT cd f(o) n=o n! n=o K n! as K ranges over the compact subsets of E. A seminorm p on HN(E) is said to be 'ITw continuous i f there exists a compact subset K of E such that for every open subset
V of E containing K there exists c(V»o such that p(f)
~
c(Y) 'ITy(f) for every f in HN(E).
We let HN(O) = ~ (H~(V),'ITy) and call this the space of OEY,open convex balanced nuclear holomorphic germs at O.
It is immediate that 'IT
o
H%morphic functions on balanced sets
137
ultrabornological locally convex space.
Let E be a locally conVex space
Proposition 3.49
{6>NcnE),TI}oo is an,5decomposition and an'&absolute w n=o decomposition for (HN(E) ,TI w) and HN(O). {(p N(n E) 'TIo}~=o is
an ,.8 decomposition and an
J
absolute decomposition for
(HN(E) ,TI ) o We leave the proof of this proposition to the reader. It proceeds in a similar manner to that used for the TO and topologies on H(U;F)
(theorem 3.19 and proposition 3.18)
also uses the following equality;
T
W
and
if B is a subset of E,
a
is a positive real number and n is a positive integer then TIaB(P)
= an TIB(P)
Proposition 3.50 transform~
for all P
E:
£P N(n E).
Let E be a locally convex space.
B, is a linear isomorphism from
The Borel
(HN(E) ,TIo) , onto
H(O(E' ,TO)) and under this isomorphism equicontinuous subsets
of CHNCE),TI o )' correspond to subsets of H(O(E' ,TO)) which are defined and bounded on some neighbourhood of zero in (E' ,TO)' Proof,
Since {(c?N(n E), TIo) }~=o is an4 absolute decomposition
(HN(E),TI ) proposition 3.13 implies that {(~N(nE),TI )6}00 o 0 ~ n=o is anA5absolute decomposition for (HN(E),TIo)S' Hence any'
for
T T
(HN(E),TI o )' can be represented as fi:o Tn where
E:
n
E
We define BT by the formula
((p N CnE) , TI 0 )' for all n.
BT = ~:o BTn'
By proposition 1.47 BTn
E /p(n(E',T o )) for Morever if K is a compact subset
every nonnegative integer n. of E and c
>0
are such that
IT(f)1
~ CTIK(f)
HN(E)
then the proof of propositionl.47 O for all n. Since K is a neighbourhood oo ~oo (BT ) < cE 1 n = 2c < 00 this ~=o TI~KO n n=o 2 and the image by B of an equicontinuous is a bounded subset of Hoo(V) Also,
for every f
in
shows that
TIKo(BT n )
of zero in implies BT
(E',T ) O E H(O)
~
and
subset of (HN(E),TIo)'
for some neighbourhood V of 0 in
since the Borel transform is an isomorphism
on each space of nhomogeneous nuclear polynomials, we have shown that B is a well defined injective linear mapping. Conversely let A ;: {g
E
o Hoo(K );
~:o
An lid
;f O) ~KO
~
I} where K is
c
Chapter 3
138
a compact subset of E.
For each g in A there exists, by
proposition 1.47, a unique sequence of linear functionals,
n
CT ) 00 T n n=o' n nand then
e: WNC E),'11 ) ' ,
o
ITnCP) I 2. c'llK(P Z;OO
n=o
IT
'n
(d f(o))1 n n!
such that BT
~NCnE).
for every P in <

c ~oo n=o
An n = d g(o); n., for all If f e: HN(E)
'n
'II
K
(d f(o)) and ~oo Tn e: (HN(E) ,lTo)'. n! n=o
Now BC~oo T) = g and so the Borel transform is a biJoective n=o n linear mapping from (HN(E)'lT o )' onto H(OCE',T ))' Since the O above also shows that A is the image under B of an equicontinuous subset of (HN(E),lT )' we have completed the proof. o
Let E be a locally convex space and let V be a finitely open subset of E' containing the origin. 'n
H CV) = {f = Z;OO d f(o) s n=o n!
e: H (V)
Z;OO
II
d f~o)1I
If
T
is a locally convex
G ; n=o
equicontinuous subset A of V}.
We let
'n
n.
A
<
00
for every
topology on E' We let Hs(OCE',T)) denote the space of germs arising from the usual equivalence relationship in
y
H
s (V),
V
ranging over at t convex balanced neighbourhoods
of zero in CE' ,T)' Proposition 3.51
Let E be a locally convex space.
Borel transform is a linear isomorphism from
The
(HN(E),lT )' onto
w
Hs(O(E',T )) and under this isomorphism equicontinuous subsets o of (HN(E),lT w)' correspond to subsets of H~(O(E"TO)) which are
defined and bounded on the equicontinuous subsets of some neighbourhood of zero in (E' ,TO)' Proof
Use the same method as in the preceding proposition
and the estimates given in proposition 1.48.
Note that as V
ranges over the convex balanced open neighbourhoods of the o compact subset K of E,V ranges over the closed equicontinuous subsets of the interior of KO. The above propositions yield a number of interesting corollaries since two topologies on a locally convex space are equal if and only if they have the same dual and define the same equicontinuous subsets in this dual. Corollary 3052
If E is a Frechet MonteZ space then
139
Holomorphic functions on balanced sets
Corollary 3.53
If E is a fully nuclear space then
T = 1T 1T T on H(E) i f H~(V) = H(V) and T bounded o 0 w w s 0 subsets of H(V) are locally bounded for each open subset V
of ES' Proof
If our hypotheses are satisfied then propositions 3.50
and 3.51 imply that 1To = 1Tw on HN(E).
By using the estimate
of proposition 1.41 we see that TO = 1To on HN(E). TO
~
~
TW
1Tw on HN(E)
for any locally convex space E it fOllows
that all of these topologies agree on HN(E). 1.41 HN(E)
Since
is a dense subspace of H(E)
topologies agree on H(E).
By proposition
and hence all of these
This completes the proof.
As a consequence of corollary 3.53 we recover a number of results previously proved by other methods; e.g. TO = Tw on (example 2.47) and T r T on o w when E is an infinite dimensional Frechet nuclear space
H(E) where E is a ~lQspace H(ExE
S)
(example 2.49)
and also obtain the following new result.
Corollary 3.54
If E is a Frechet nuclear space then T
T
o
W
on H(E) . Theorem 3.55
property.
Let E be a Frechet space with the approximation
The Borel transform is a linear topological
isomorphism from CH(E),TO)S onto
HN(O(E',T ))' O
Since {(~(nE),To)S}:=o is a Schauder decomposition for
Proof
(H(E),T)~ (corollary 3.14 and theorem 3.19) any T £ (H(E),T)' o ~ 00 0 n can be represented as I: T where T £ c6>C E),T )'. By proposn=o n n 0
ition 1.61 BTn £ !PNCn(E',T )) for every nonnegative integer o n. If L is a compact subset of E and c> 0 are such that
ITCf)
I
~ c
IIf~L
for every f in HCE)
then, by proposition
1.60,there exists K compact such that 1TKoCBTn) ~ c for all n. O Since K is a neighbourhood of zero in (E',T ) and o ~:o 1T~KOCBTn) ~ c Tn = 2c < 00 this implies that BT£ HN(O)
ti:o
and the image by B of an equicontinuous subset of CHCE),T )' 00
is a bounded subset of HN(V) (E',T
O
0
for some neighbourhood V of 0 in
)'
Conversely let A = {CPn):=o£ HN (O);1i:o 1T KO (P ) ~ I} where K n
Chapter 3
140
is a convex balanced compact subset of E.
If (Pn)~=o
E
A
then for each nonnegative integer n there exists a unique E: (p(n E), T ) I such that BT = P and n o n n every P in ~(nE). Since 'n An oo
T
L'"
n=o
IT
(LU~J)I
n
n!
< L

n=o
lid
f~o)"
n.
I Tn (P) I .s.
lip 11K for
K
oo
it follows that T = L T dH(E) ,T ) I and BT E: A. This also n=o n 0 shows that A is the image of an equicontinuous subset of Since {(a'(n E ) ,T )}oo is an J decomposition for o n=o (H(U),T ) propositions 1.61 and 3.13 imply that O (H(E) ,TO)
I.
o n (E ,T )),1I )}n=o 00 {((TN( is an"," decomposition for (H(E),TO)S' ' o w Since (H(E) ,TO) is a semiMontel space its strong dual
(H(E),TO)S is a barrelled space. {/PNCn(EI ,fa)) .
,11}
W
Hence, by proposition 3.49,
is anJ decomposition for two barrelled
n
topologies on HN(O)
and proposition 3.12 says that these
topologies must coincide. ally and topologically.
Hence
(H(E) ,TO)e, '" HN(O)
algebraic
This completes the proof.
Let E be a quasicomplete nuclear and dual
Lemma 3.56
nuclear space.
Then HN (0) = H (0)
(algebraically and
tapa logical ly). ~v e
a 1 way s h a v e HN (0) C H ( 0) .
~(nE) T
W
11
W
By the 0 rem 1. 2 7
for each nonnegative integer n and moreover
on ~(nE) by proposition 1.44.
Again by proposition
1.44 we can choose for every neighbourhood W of zero another neighbourhood V of zero such that L'"
n=o Iience HN(O)
= H(O)
11
(P ) < ):00 W n  n=o
algebraically and topologically.
that a subset of HN(O)
Note also
is contained and bounded in some HN(V)
if and only if it is contained and bounded in some H"'(W) , V and W being neighbourhoods of 0 in E.
If E is a Frechet nuclear space then
Corollary 3.57 (H(E),TO)S ~ H(OE
f3'
).
Holomorphic functions on balanced sets If E is a Frechet nuclear space then
Corollary 3.58 T
TO on H(Ej i f and only i f lim,
o
141
(Hoo(V),
II
!IV) is a regular
V~O,
V open inductive limit where OEES'
Proof
Since E is a Frechet space TO
=
TO on H(E) if and only
if (H(E),T O) is an infrabarrelled locally convex space. A locally convex space is infrabarrelled if and only if strongly bounded subsets of the dual are equicontinuous. (H(E) ,TO)'S
~
Now
H(OE') and the equicontinuous subsets of
(H(E),T O)' are th~ subsets of H(O) which are contained and bounded in Hoo(V) for some neighbourhood V of zero in F' (3
So (H(E) ,TO) is infrabarrelled if and only if each bounded subset of H(O) is contained and bounded in Hoo(V) for some neighbourhood V of zero,
.!l..!!!..... o EV,
(Hoo(V),
II Ilv)
i.e. if and only if
is a regular inductive limit.
V open
VCES
Example 3.59
Example 3.47 shows that H(O) is not a regular
inductive limit if 0 that TO
r
E
Q;(N).
TO on H(~ N).
By corollary 3.58 this shows
We have already obtained this result
by a different method (example 2.52).
Example 2.52 and
proposition 3.58 also show that if E is a Frechet nuclear space which does not admit a continuous norm then oo 1 im., (H (V) ,II Ilv)
o EV,
V open veE' S
is not a regular inductive limit. §
3.4 SEMIREFLEXIVITY AND NUCLEARITY FOR SPACES OF HOLOMORPHIC FUNCTIONS Our first result follows from a general theorem concern
ing semireflexive locally convex spaces which have equiSchauder decompositions. Proposition 3.60
Let U be a balanced open subset of a
locally convex space E, let F be a Banach space and let
Chapter 3
142 T£{T o ,T 0, b'
Then (H(U;F),T) is
Tw ,T w, b' T,,} on H(U;F). \J
semireflexive i f and only i f (~(nE;F) ,T) is semireflexive for each nonnegative integer nand H(U;F) is T.S. Tcomplete. In particular this means that reflexive if and only if (H(U;F),T Corollary
3.61
w.
(H(U;F).T b)
W
)
is semi
is reflexive.
If U is a balanced open subset of a
Frechet space and F is a Banach space then (H(U;F) ,T), where T£{T o , T W , T~}, is semireflexive i f and only i f . (f(nE;F) ,T) U is semireflexive for each nonnegative integer n. We now show that
(H(U) ,TO)
is a nuclear space if U is
an arbitrary open subset of a quasicomplete dual nuclear space.
Since the projective limit of nuclear spaces is
nuclear and the compact open topology is a local topology it suffices to prove this result for convex balanced open sets. This result can easily be proved for entire functions by using Taylor series expansions and the following lemma whose proof is already contained in the proof of theorem 1.43. n
We let
Note that s is finite by Stirling's
s = sup 1/ . n (n!) n formula.
Let K and (xn):=l be respectively a compact
Lemma 3.62
balanced subset and a compact sequence in the locally convex space E.
Let L denote the closed convex balanced hull of
(sxn):=l'
If (An)~=l £ tl and II¢II K '::'~:II"nl IHxn)1 for every ¢ in E' then for each positive integer m there exist a sequence in iI'
0m,n)~=I'
such that Proof ilL 11Km
~:l
with
lip 11K .::.
I"m.nl
~:I I"m,nl
=
IA n
. 1
n l ,· .• n m=1 for every L in ,Ls (IDE).
I"nl)m, and
(x~)~=l
CL
Ip(x~) I for every P £!PCl!lE).
By induction one sees
,::,,L.
C~:l
An
m
Cas in theorem 1.43) I
I L (x
n
, 1
. 'Xn ) I m
An application of the polarization
143
Holomorphic functions on balanced sets
formula easily completes the proof.
A sequence (xn)n in a locally convex space is said to be rapidly decreasing if nPx
,
n
as n
0
, ~ for every
positive integer p. Lemma 3.63
Let U be a convex balanced open subset of a
quasicomplete dual nuclear space E and let K be a (convex balanced) compact subset of U. There exist a finite dimensional subspace F of E, Ko a compact subset of F, Uo an open subset of F, (An):=l Eel' (xn):=l a rapidly decreasing sequence in E and W a neighbourhood of zero in E such that ex>
Ko C Uo ' h=l K CK
ex>
0
CU 0
eu 0
Proof
I
IAn l 5 2s and
+ {h=l fl x ; Iflnl n n + {h:l fl x ; Iflnl n n ex> + {h=l fl x ; Iflnl n n
< < <
I An I for aU n} I
for aU n}
I
for aU n} + weu.
Let V denote a convex balanced neighbourhood of zero
in E such that K + VC:U.
Since any compact subset of a
quasicomplete dual nuclear space is contained in the convex hull of a rapidly decreasing sequence we can choose {y}
n n
rapidly decreasing sequence in E whose closed convex hull contains K.
Choose N a positive integer such that
4
lOan YnEV for all n
>
N where a
=
2 11
sIb·
Let F denote the subspace of E spanned by {Y l ' {~~l any n ; 3 (an):=N+I such that
and let Ko ex>
ex>
I a n I 5 1 and Ii=l anY n EK}. Let By our construction K is a compact subset of F. 0 I U is an open subset of F and K CU . U = K + 'l(V ('I F) . o 0 0 0 0
;;=1
Let A n
1 2a(N+n)2
integer n.
2 2a(N+n) YN+n for each positive 2 I I I 11 I I An I 5 2a Ii=l 6" 2s and 2 2a n
and x ex>
Since Ii=l
n
~
a
144
Chapter 3
K eKo + {hN K
0
CI.
nYn; fl>Nlanl a
co +{ff=l
CK 0
+q~:l
2a(N+n)Z
ISn l
::.
ISn l
::.
co
C U0 +{fl= I Sn x n'. co
I
5 fl=l
+ I V + I Ii +
CK
4
4
Ia n I
::.
1}
ISn l < I An I for all n}
S x n n
I C K0 + 4 V +
1}
co 2 Za(N+n) Y + ; L I N n
n
"" CK 0 +{ff=l Sn x n
<
I n
Z
I for all n} I for all n} +IT V I V Ii + IT
1 V + I VCK + VCU 3 IT
this completes the proof if we let W
I
IT
V.
We now prove the main result of this section.
We shall
assume that (H(U), TO) is a Frechet nuclear space if U is an open subset of ~n.
This is a well known finite dimensional
result and is given in a number of books on functional analysis.
It is also a special case of a result which will
be proved,
independently of the following result, in
chapter 5. Theorem 3.64 Then
Let U be a quasicomplete dual nuclear space.
(H(U),T ) is a nuclear space i f U is an open subset of E. O
Proof
We may suppose that U is convex and balanced.
Let K
be a compact subset of U and let Ko , U0 ,(A n )'" n= I' CXn)nco __ I and W have the same meaning with respect to K and U as in lemma 3.63.
By the nuclearity of (H(UO),T ) we can find O (Sn)~=l £t l and (
(H(U ), o
7
0
such that
) I
IlfilK
::. fi:l1f3nl o
On multiplying each f3
I
by a constant if necessary we may n suppose that there exists a relatively compact subset KI of Uo '
145
Holomorphic functions on balanced sets
Ko
c: KI
H(U
O
,
I
such that
for all n and every f
in
I
)'
Let K2 = {nL:_I en xn; Since
2
I
2s
IIfllK2 for each f ~
where L I n=
IA n I for all n}.
I An I I
~ ~:l
11
Ie n I <
for every
and
lemma 3.62 implies that Am
~ I f(o) I
+ ill:l fi:l
I am,n
I I d ~io)
(xm,n)
I
defined and hOlomorphic on a neighbourhood of K2 Ia
m,n
I
I < 
(2)
m
~
for all m and ( x ) is contained m,n m,n=l K . 3
KI + K3 C
in E I
U + K3 + Wand hence KI o
Note that
+ K3 is a compact subset
of U. If yE K3 + W then fy defined by fy(x)=f(x+ y )
Now let f E H(U).
is an element of H(U )' o
¢'co ;
Y EK3 + W
If
+l
E H' (U ) o
E
q;
Since L~ n=o
is an element of H(K +W). 3
then the function
An d ~ix) (y)
= f(x+y)
for all x in U and y in K3 + Wand the series converges o uniformly with respect to x over the compact subsets of U .it o follows that
1Cfy )
dn¢Cf ) (0) (y) n!
and
for all y in
E.
Hence
IIf 11K
<
<
~ <
(dnf(.) (y) ) n! An
~
= L n=o
l:~
n=l
sup YEK fi:l
a
+K
2
fi:l
l f3 n l
I
2 l f3 n l sup YEK
l f3 n l (l
I
+ ffi:l
for all YEK +W 3 (y))
146
Chapter 3
Since 1:"" n=l
Ak
))\ ( d f~')(x k. m,n .::. .::.
I f(x+y) I
sup
xEK l YEK3
II f Il
K.:
where
~:l
°n
nl::l
of (H(U),T
on
O
for all n,m and k we have shown
!ljin(f)!
<
00
and (Iji)"" is an equicontinuous subset n n=l
This completes the proof.
)"
If U is an open subset of a is a Fpechet nucleap space.
Corollary 3.65 (H(U),T
3.5
O
)
~1~space then
EXERCISES
3.66*
A convex balanced absorbing subset of a vector space
is called a barrel.
A locally convex topOlogy T is said to
be dbarrelled (resp. dinfrabarrelled)
if every barrel
(resp. every barrel which absorbs bounded sets) which is a countable intersection of neighbourhoods of zero is a neighbourhood of zero.
Show that the collection of all d
barrelled (resp. dinfrabarrelled) Let T
dt
topologies form a Qfamily.
and Tdi denote the dbarrelled and dinfrabarrelled
topologies associated with T respectively.
If U is a balanced
open subset of a locally COnvex space E and F is a Banach space show that TO
Let
= Tw,dt and
TW b=T ,
0,),
d'1 on H(U;F).
(E.T) be a locally convex space.
Le t Tl = T.
Suppose ta has been defined for all a strictly less than the ordinal number a
o
'
Let
(E,T )= lim (E,T a
+
(lo
a
a
).
Let T
CI.
denote 0
147
Holomorphic functions on balanced sets
the topology on E inherited from ((E,T a )'eJ'i3. o there exists an ordinal number Show that eE, T Yo convex space and that
T
associated with T. 3.68
Yo
for all Yo ) is an infrabarrelled locally y
such that T
Show that =
y
T
is the infrabarrelled topology
Let (E,T) be a locally convex space which contains
a countable fundamental system of bounded sets.
Show that
(E,T di ) is a DF space. 3.69*
Let
i ,
q p
endowed with the
1 .::.. p .::.. q < + 00,
ret
norm. if and only if p = q. 3.70*
denote the
tp space
Show that qip is a barrelled space
Let {En}n be a Schauder decomposition for the
locally convex space E and let (Pa)aEr be a fundamental system of seminorms of E. LetT' denote the locally convex topology generated by the seminorms qa(x) = sgP Pa(un(x)) as a ranges over r. Show that {E} is an equiSchauder decomposition for (E,T') and that n n
T' is the weakest locally convex topology on E, finer than T, which coincides with T on each En and has {En}n as an equiSchauder decomposition. 3.71
If {E}
n n
is a Schauder decomposition for (E,T ) and l
(E,T 2 ) show that Tl and T2 have the same associated barrelled topology. 3.72
Let (E,T) be a locally convex space,
is weak* quasicomplete if and only if (E,T)' 3.73*
Show that E'
=
(E,T )'. t
Let {En}n be a shrinking decomposition for the locally
convex space E.
Show that {(E )'i3}
decomposition for ES.
n
n
is an equiSchauder
Chapter 3
148 3.74
Show that {Q>Hy(nE;F) ,TO}:=O is antS' decomposition
and an iabsolute decomposition for
(HHy(U;F)~o)
whenever U
is a balanced open subset of a locally convex space E and F is a quasicomplete locally convex space. 3.75*
If U is an open balanced subset of a locally convex .on
iOn
OJ
00
space E show that {Ir( E),To,t}n=o and {If ( E),To,ub}n=o are ,j'decompositions and ,gabsolute decompositions for (H(U) ,T
0,
3.76*
t)
and
(H(U) ,T
o,u
b)
respectively.
If E is a DF space show thatT
o,t
for any positive integer n and hence deduce that T on H(U) 3.77* TO
=
=
T
0
for any balanced open subset U of E. If E is a Frechet space and F = (E', TO)
TO
o,t
show that
on H (F) .
3.78*
If E is an infinite dimensional reflexive Banach space
and F
(E,a(E,E'))
3.79
Show that H(E)
show that
is T.S.
((?(IF),Tw)'
~ (E')* and that
TW complete but not T.S.
T
o
complete when E = [N x [(N). 3.80*
Let T
p
denote the topology of pointwise convergence.
If E and F are locally convex spaces such that H(E) are T.S.
T
p
and H(F)
complete and separately continuous polynomials
on E x F are continuous show that separately continuous holomorphic functions on E x Fare holomorphic if and only if H(E x F) 3.81
is T.S.
T
p
complete.
If E is a quasicomplete infrabarrelled space show
that (H(E),T ) is Frechet if and only if E is the strong O dual of a Frechet Montel space. 3.82
If E is an infrabarrelled complete DF space show that
(H(E),T ) o
is barrelled if and only if E is a Montel space.
149
Holomorphic functions on balanced sets
Show that a sequentially complete barrelled space with
3.83*
a basis is complete. If E is a quasicomplete nuclear and dual nuclear
3.84*
space show that the compact open topology on H(E)
is
generated by all seminorms of the form p
An
p:'"
Z '"
d f(o)) K n=o n!
An
(d f(o)
'IT
n=o
K
n!
as K ranges over the compact subsets of E.
3.85*
Let 0 be the origin in the Banach space E and let B
be the unit ball of E.
Show that the topology on H(O)
is
generated by the seminorms p (f)
where
(a )
Inn
I a n I In
'+
=
Z'"
n=o
Ia n I
ranges over all sequences of scalRrs such that o
as n 
"'.
3.86*
Let 0 be the origin in [(N),
of H(O)
is generated by all
where la
(a)
n n
show that the topology
seminorms of the form
ranges over all sequences such that
Il~ ~
0
as n +
~
and K ranges over the compact
su~sets
of [(N).
3.87
If {fn}n is a sequence in H(O), 0 the origin in
and if,
for each n,
there exists a neighbourhood of zero
Un in (n such that fmlu
= a all m >
n show that {fn}n
n
is a very strongly convergent sequence in H(O).
3.88*
Let U be a balanced open subset of a quasicomplete
locally convex space E. let K = {xEE;
I p (x)
For each compact subset K of E
I ~ lip 11K for every P in (J> (E)).
150
Chapter 3 A
Let
U =Jiu R
If f
=
Show that U is a balanced open subset of E.
K compact co dnf(o) ~=o n! 
E H(U)
l:co
converges and
show that n=o A
rJ
and defines a hOlomorphic function f on U. mapping f
f
E (H(U) ,TO) 
isomorphism.
E:
A
(H(U) ,TO)
If To = Tw on H(U)
(the boundary of " U in E) there exists an f 3.89*
'"
and
~
is a linear
E U,
such that
~
+
'" n
f(~
n
)+
~
as n co
let
Ilpl~ = TIB(P).
show that p.Q E
3.90*
fi)
~N(
n+m
E)
E
au
+
as n
Let E be a Banach space with unit ball B.
P dPN(n E )
~
show that for each
n
in H(U)
Show that the
co
+
co.
If
and Q E !PN(m E)
I f P E(?N(n E)
and
If U is a balanced open subset of a metrizable
locally convex space show that TO = To,t on H(U). 3.91*
By using the fact that there exist discontinuous
polynomials on ([I,
I uncountable,
show that
(H(([I) ,TO)
is not a semiMontel space. 3.92*
Show that
lim
>
n
Let E and F be locally convex spaces and let U be a
3.93
balanced open subset of E. TE:{T
Show that
(H (U; F) ,T) ,
,
T , T r } , is semiMontel if and only if (H(U;F) ,T) is o w u T.S. T complete and C&CnE;F),T) is semiMontel for each nonnegative integer n.
3.94
Let E be a locally convex space.
Show that
(H(U) ,TO)
is complete for every open subset U of E if and only if (~(nE),T ) is complete for each nonnegative integer nand o is T.S. TO complete for each convex balanced open
H(V)
subset V of E.
151
Holomorphic functions on balanced sets 3.95* that E
n
Let {E}
be a sequence of Banach spaces.
n n
(H(~"'l E ) n=
n
,T
0
Show
is a semiMontel space if and only if each
)
is finite dimensional.
3.96*
If E is a locally convex space and f£HNCE)
dnf(x) £iPNCnE)
show that
for every x in E and every positive integer n.
Show, by counterexample, that the above condition on f£H(E)
is not sufficient to insure that it lies in HN(E).
Show also that HN(E)
is a translation invariant subalgebra
of H(E). 3.97*
Let E be a locally convex
linear space.
spac~
A function f £ HCE;F)
and F a normed
is said to be of
exponential type if there exist a continuous seminorm a on E and positive numbers c, for every x in E.
Let
C such that
Ilf(x)
II
~ C exp
(ca(x))
Exp(E;F) denote the set of all
holomorphic functions of exponential type from E into F. Show that f = ~'" n=o
dnf(o) n1
£
Exp(E;F)
if and only if there
exists a continuous seminorm a on E such that lim sup [ sup
{II
A
dnf(o) (x)
II;
1/
a(x) ::.. I}] In < "'.
n + '"
3.98
If E is a Banach space and f£ H(E)
f £ Exp(E;C)
=
Exp(E)
show that
if and only if the restriction of f to
each one dimensional subspace of E is a function of exponential type. 3.99
If E is a locally convex space show that the
mapping
An f = ~'" d ff o ) n=o n.
£ Exp(E) +l
!;'"
n=o
is a linear bijection. Using the above, or otherwise, describe a locally convex topology on Exp(E)
so that the above bijection is a linear
Chapter 3
152
topological isomorphism. Let E be a Banach space and let f and g be hOlomorphic
3.100
functions of exponential type on E. function on E show that h
E
If h
Let E be a locally convex space.
3.101
= fig
is an entire
Exp(E). An element f of
HN(E) is said to be of nuclear exponential type if there exists a convex balanced open subset V of E such that An
lim sup TIv(d f(o)) n +
lfn <
00.
00
Let EXPN(E) denote the space of all holomorphic functions of nuclear exponential type on E.
Show that the mapping

f
is a linear bijection. 3.102*
Let V and U be open subsets of the locally convex
spaces E and F respectively.
Let TI be a continuous linear
mapping from E into F such that TI(V) is a compact subset of U.
Let RjHOOCU)   Hoo(V) be defined by RCf)
Show that R is a compact mapping. that HCK) is a
:bJg
=
fOTIiv'
Using this result show
space whenever K is a compact subset of
a FrechetSchwartz space. 3.103*
If U is an open subset of a locally convex space E
and F is a semiMontel space show that(HHy(UjF) "0) is a semiMontel space. 3.104 *
that
If E is a quasicomplete dual Schwartz space show
(H(U) "0) is a Schwartz space for any open subset
U of E.
153
Holomorphic functions on balanced sets
NOTES AND REMARKS
§3.6
The concept of Qfamily (definition 3.1) J.
Schmets
of J.
[627]
Schmets
is due to
(see also chapter 2 of the lecture notes
[628]
and Ph.
Noverraz
[553,556]) and
developed naturally from the results of earlier authors on particular associated topologies.
Y.
Komura
first to discuss associated topologies.
was the
[394]
He was interested
only in the associated barrelled topology and proved An alternative proof of this proposition
proposition 3.5.
using the axiom of choice in place of transfinite induction is due to M. A.
Roberts
topology.
[608]
Kennedy (Lecture,Dublin,
December 1979).
also studies the associated barrelled
The corresponding results for the infrabarrelled
topology and for the quasicomplete and sequentially complete cases
(proposition 3.6)
are due to K.
studied by H.
Buchwalter in
topology by K.
Noureddine in
[108], the barrelledbornological [533]
ainfrabarrelled topologies by K.
and the abarrelled and Noureddine and J.
Schmets
General results for Qfamilies are given in
[535].
J.
Noureddine
The associated ultrabornological topology is
[532].
Schmets
[627,628].
introduced by H.
Kelley spaces
Buchwalter [107]
(example 3.4) were
(see also K.
Noureddine
[534]).
Schauder decompositions of Banach spaces were first defined by M.M.
Grinblyum [283]
(see B.L.
Sanders
and extended to linear topological spaces by C.W. and J.R.
Retherford [483].
[624])
McArthur
The only result we use without
proof (in the proof of proposition 3.60)
is due to
B. L.
Sanders
We refer to
N.J.
Kalton
[624]
and T.A.
[370,371]
Cook
[168].
for further details.
The concepts of absolute decomposition, 4decomposition and4absolute decomposition are new and are introduced here as a suitable technique for treating hOlomorphic functions on balanced open sets.
Propositions 3.10, 3.11 and
Chapter 3
154
3.13 are new while a stronger form of proposition 3.12 may be found in Ph.Noverraz
[553,556].
Schauder decompositions were introduced into infinite dimensional holomorphy by S.
Dineen and all the
results of §3.2 and a number of those in §3.3 are to be found in [185].
These results were motivated by earlier
results concerning holomorphic functions on Banach spaces eS.
Dineen
[177], R.
Aron
[17]).
The arrangement of the
material is, however, new and more coherent than that given in
[185]. The application of associated topologies, in
conjunction with Schauder decompositions,
to the study of
holomorphic functions on locally convex spaces is due to Ph.
Noverraz
[553,556]
where he proves theorem 3.26 and
proposition 3.29.
Propositions 3.29, 3.30, 3.31 are given
in S.
and lemma 3.28 is due to L.
[509] .
Dineen
[185]
Nachbin
In view of theorem 3.26 we may ask if To,t
TO on H(U),
U a balanced open subset of a locally convex space. Proposition 3.41, exercise 3.76and corollary 5.26 all give a positive answer for special cases but Y.
Komura's
[395]
example of a noncomplete Montel space shows that we may have To,t
t
TO even on E'.
Recently J.M.
Ansemil
Ponte [10] have shown that T t Tw on (pelE), o ,ub E an infinite dimensional reflexive Banach space with the
and 5.
weak topology,
and hence we do not,
in general, have
To,ub = TO on H(U). The completeness of (H(U;F) ,Tw) has been investigated by many authors and the result presented here 3.44) may be found in S.
Dineen [200].
in a series of results which appear in S. R.
Aron [171, 5.B.
P.
Aviles and J.
Dineen [177,185],
Chae [1201, J. Mujica [503]
Mujica [41].
(corollary
It is the latest and
Aspects of the completeness
question will arise in each of the remaining chapters. Taylor series completeness was introduced by 5.
Dineen [185].
155
Holomorphic functions on balanced sets Lemma 3.33 is new and a general result of the same kind for ~absolute
decompositions can easily be stated and proved.
Corollary 3.34 and proposition 3.35 are new. is given in S.
Proposition 3.36
'w
Dineen [185] where one may also find a
analogue of proposition 3.36. The classical Montel theorem says that closed bounded subsets of (H(U)" compact. result
o
)
(U an open subset of I[n)
are
A number of different generalizations of this
(known collectively as Montel theorems)
for
holomorphic functions of infinitely many variables have appeared in the literature.
The variety of results are
obtained by varying the underlying locally convex spaces, the concept of differentiability and the topology on the corresponding space of holomorphic functions.
Most of the
proofs require an application of Ascoli's theorem. first result of this kind is due to D.
Pisanelli
The
[571]
for
JJJ,g spaces and this is a particular case of corollary 3.38. Further Montel theorems are to be found in D. [576,578,582], D. [149]
Lazet [423], J.F.
Pisanelli
Colombeau and D.
Lazet
(this article contains proposition 3.37 and
corollaries 3.38 and 3.39), J.F. S. Dineen [185,194]
Colombeau [141]
and
(see also exercises 2.84 and 3.103).
A
number of the above authors also prove infinite dimensional versions of the classical Vitali theorem. M.C. Matos
[462] discusses locally convex spaces which satisfy
a "Montel" property and shows that they are related to locally convex spaces which satisfy the conclusion of Zorn's (theor~m
theorem of
'w
2.28).
Ascoli style characterizations
compact sets are due to L.
[120], R.
Nachbin [509] ,S.B.
Aron [17], M.e. Matos [461]
and J.A.
Chae
Barroso
[47,48] . Propositions 3.40, 3.41, 3.45 and corollaries 3.42, 3.43, 3.44, and 3.46 are due to S.
Dineen [200].
Example
3.47 is new and related to an example of R. Aron given in R.L. Soraggi
[664].
156
Chapter 3 Analytic functionals on locally convex spaces
are usually represented either as functions of exponential type or as holomorphic germs at the origin. natural
linear topological
(but unfortunately not algebraic)
isomorphism between these representations 3.93,
There is a
(see exercises
3.94 and 3.95). The exponential type representation is useful in
studying convolution operators
(see appendix I)
while we
have found the germ approach useful when investigating topological properties of holomorphic functions.
Since the
results here on analytic functionals were originally proved using Taylor series expansions about the origin we are essentially using the original method.
Sometimes
however the Schauder decomposition approach can be more efficient
 as in theorem 3.55.
Holomorphic functions of nuclear type 3.48) were introduced by C.P. L.
Nachbin
[511].
The first
(definition
Gupta [295,296]
and
infinite dimensional
representation theorem for analytic functionals by holomorphic germs is due to P.J. Boland [85] who proved that o (HN(U) ,ITo)S ~ H(U ) whenever U is a convex balanced open subset of a
J)JIl
space.
This is a stronger result than
proposition 3.50 forJlJl[spaces.
Propositions 3.49,
3.50,
3.51 and corollary 3.52 are new.
Corollary 3.54 is due
to P.J.
and an alternative proof
Boland and S.
is given in §5.4.
Dineen
[90]
Theorem 3.55 is due to R.
Ryan
[620]
Lemma 3.56 is new while corollaries 3.53 and 3.57 are proved in P.J.
Boland and S.
Dineen
[90]
assumption that E has a Schauder basis. and example 3.59 may be found in P.J. [91]
and S.
Dineen
[202].
under the additional Corollary 3.58
Boland and S.
Dineen
Further representation theorems
for analytic functionals on a Banach space are due to J.M.
Isidro
[351]
while the classical theory for functions
oro n e com pie x va ria b l e i s due to A. G.
Kothe
[396]
and C.L.
da Silva Dias
Grot hen die c k [661].
A.
[ 2 8 5] ,
Martineau
157
Holomorphic functions on balanced sets
[451]
investigates the case of several variables. Proposition 3.60 and corollary 3.61 are due to
S.
Dineen [185].
Theorem 3.64 was first proved for entire
functions on quasicomplete nuclear spaces by P. ~elimarkka
see also E.
[526],
independently, to arbitrary open sets by P.J. and L. by L.
Waelbroeck [713].
Perrot
[160,161]
Boland [86]
Our proof is close to that given
Waelbroeck, who also proves lemma 3.63.
proof of theorem 3.64 is given by J.F. B.
Boland [83],
and afterwards extended,
A further
Colombeau and
and for fully nuclear spaces with a
basis we provide an alternative proof in chapter 5 (corollary 5.22).
Applications of theorem 3.64 to lifting
theorems for linear mappings are to be found in W. [363] B.
Kaballo
and to the classification of Stein algebras in
Kramm [398,399]. Extensions of theorem 3.64 to A and s nuclearity and
to nuclear bornologies are given in K.D. B.
Gramsch and R.
Bierstedt,
[67], K.D.
Bierstedt and R.
~Ieise
[713], J.F.
Colombeau and R.
Meise
Meise
[69,70], L.
Waelbroeck
[152]' J.F.
Colombeau and B.
Perrot
[157,159,160,161,165].
For example the following result is proved in [152]; let E be a quasicomplete locally convex space, then (H(U;F),T
O
~s
)
an s nuclear space for any open subset U of E
if and only if (E Sat z 1. 12
0
f
I
,T )
o
and F are both s nuclear spaces
(see
[67]).
An approach to the mathematical foundations of quantum field theory using nuclearity and infinite dimensional ho1omorphy is given in P. 411, B.
413,414,415,416,417]
Perrot
[158]
and J.F.
exercise 3.104.
[406,407,408,409,
Colombeau and
Co1ombeau [145)).
Corollary 3.65 is due to P. Schwartz property for
Kree
(see also J.F.
(H(U) ,T)
Boland [82,83].
The
is discussed in our notes on
This Page Intentionally Left Blank
Chapter 4
HOLOMORPHIC FUNCTIONS ON BANACH SPACES
Banach spaces and nuclear spaces play an important role in linear functional analysis and also in classical analysis by way of application.
This chapter is devoted to the study of
holomorphic mappings between Banach spaces and in chapter 5 we discuss holomorphic functions on nuclear spaces.
As one
would expect, since every nuclear Banach space is finite . dimensional,
these two topics proceed along quite different
lines but both confirm that infinite dimensional holomorphy leads to concepts and results which are of interest in themselves and quite different from what one would expect from the underlying fields. In this chapter we find that there is a rich interaction between the theory of holomorphic functions and the geometry
of Banach spaces.
By the geometry of Banach spaces, a topic
that has undergone rapid development in the last fifteen years, we mean the study of geometric properties of the unit ball such as smoothness, the existence of extreme points, dent
abitity, uniform convexity, sequentiat compactness etc. If E is a Banach space then the compact open topOlogy on H (E)
is genera ted by
subsets of E.
II I
k as
K ranges over the compact
Our motivating problem is the following;
there exist any other seminorms on H(E) II Ik for some subset A of E?
do
which have the form
If such a seminorm
II
IIA exists
it will always be '0 continuous and if A is not precompact the seminorm will not be 'w continuous.
Since
IIIIA is a semi
norm if it is finite we are tooking for nonretativety compact A such that
IlfilA <
00
for every f in H(E). 159
This problem has
Chapter 4
160
led to much of the research we report in this
In the first
section we discuss
chapter.
a few general
properties of holomorphic mappings between
Banach spaces.
Some of these are unrelated to the topological problem but are of interest in themselves. §4
.1
ANALYTIC
EQUALITIES
AND INEQUALITIES
The theory of holomorphic
functions of one or several
complex variable contains a number of interesting and useful equalities and inequalities and it is natural to extend these to infinitely many variables. interest if they satisfy at
Such generalizations are of
least one of the following
criteria; a)
they require new nontrivial proofs
(and a study of these
in turn may lead to improved and even new finite dimensional results) , b)
they lead to applications not covered by the corresponding
finite dimensional results, c)
they give rise to a classification problem for locally
convex spaces, d)
they lend themselves to new interpretations which in turn
suggest new concepts and problems
(which may even be trivial
or nonexistent in finite dimensions).
We present here extensions of three well known results from the theory of one complex variable;
Schwarz's lemma,
maximum modulus theorem and the CauchyHadamard formula. Since these extensions will not be required later we do not give a comprehensive account.
For both Schwarz's
lemma and
the maximum modulus theorem we need the concept of an extreme point.
Definition 4.1
Banach space.
Let K be a convex subset of a complex A point e of K is
the
161
Holomorphic functions on Banach spaces
( a)
a real extreme point i f {e implies x = 0,
lb)
a complex extreme point i f {e imp lies x = o.
AX;
+
+
1
<
AX;
0
A
::.
nCK
< I AI <
l}C K
It is clear that every real extreme point is a complex If every point of norm one is a real extreme
extreme point.
point of the closed unit ball of E then E is called a rotund or a strictly convex Ba~ach space. convex if 1
<
P <
00
LP(X,n,~) is strictly
for any finite measure space
(X,n,~).
If
every point of modulus 1 is a complex extreme point of the closed unit ball of E then we say E is a strictly cconvex Banach space.
Ll(O,l) is a strictly cconvex Banach space
which is not strictly convex. Now let D
=
{zd:;
Izl
<
n.
Schwarz's lemma in one
=
variable says that if f E H(D;D) and f(o)
then
0
If(z)1 ::. Izlfor all z ED and moreover if If(z )1 o
some Zo ED then fez)
=
Iz
0
I for
cz for all z in D where c is a
=
constant of modulus 1.
We use the first part of this result
to prove the following lemma, which is also useful in extending the maximum modulus theorem, and extend the second half to mappings between Banach spaces. Lemma 4.2 for all z
Proof
If fE H(D;D) then If(o) I
lIzl I
+ 
2Iz
I fez) f(o)1 ::. 1
D\{o}.
E
If I fez) I
1
=
for some zED then the one
dimensional maximum modulus theorem implies that f is a constant mapping in which case the above result is trivial. Hence we may assume fEH(D;D). Z
>
ZCl
(I I Cl
<
laz mapping z   g(z) g(o) hence
=
o. If(z)
1) =
The Mobius transformation
maps D onto D and
Cl
to o.
Hence the
fez) f(o) belongs to H(D;D) and lf(o)f(z)
By Schwarz's lemma Ig(z)1 ::. Izl  f(o)1 ::.Izl·
Ilf(o)f(z)1
for all ZE D and
all zED.
Now
Chapter 4
162
Ilf(o)tTci1+noJ (f(o) fez)) 1
Ilf(o)f(z)1
~
1lf(0)1
2
+lf(0)llf(0)£(z)1
~
If(z)f(o) I Hence
and thus
2
Izl (lIf(o) 1 )+lzllf(0) Ilf(o)f(z) I. 2 ).
(llzllf(o)I)(lf(z)f(o)I)::.. IzIOlf(0)1
Since
If(o)1
shows
(1lzl)
<
1 we have lIzl ::.. 1lzllf(0)1 If(z)f(o)1 ::.. 21z1
Olf(o)I).
and this On dividing
across by 21z1
we complete the proof.
Theorem 4.3
Let E and F be Banach spaces with open unit balls
U and V respectively.
Let f E H(U;Y) and suppose dfCa) is an
isometry from E onto F.
Then
f(x)
df(o)Cx)
=
for all
x
in U and in particular f is an isometry from U onto V. Proof
We first note that by replacing f by dfCo)
we may assume that E identity map on E. otherwise, '$ EE " g(z) Ig
Hence
s uc h t h at '$ Cf ( 0 ))
zfi£l <pof(lJf(o) II)
2
::"1lg'(0)1
for all 2
,
2 Ig(a) 1 =lIf(0) 112
and f(o) w~(x)
We first show that f(o)
~
(0)1
=
=
0
f
= I, where I is the
and df(a)
= o.
Suppose
then by the HahnBanach theorem, there exists
11'$" = 1, =
= F
1
zED.
= IIf ( 0) II.
Let
Since gEH(D;D) we get
(see example 2.31).
::"1lg'(0)12::"11¢(lIi~~~II)I~
0
Now fix ~ED'{o} and let
o.
L:J.ll 1 21~' ["f
f(~x)x]
.
for every x ln U.
To complete the proof it suffices to show, is identically zero for every F,. Our first step in proving this is to show IIx + AW~(X)" ~ IIxll for all x in U and :>..ED. x EU,{o}and
formula h(o) Since f(o)
=
By lemma 4.2,
II
0
and h(z)
and df(o)
= I
=~
Let
DeHne h by the
<pof(,;,,)
if ZED,{o}.
it follows that hEH(D;D).
letting z = ~IIxll, we have
Ho[omorphic functions on Banach spaces
163
~ Ilx II. As
was an arbitrary element of the unit ball of E'it follows
by the HahnBanach theorem, that IIx+Aw!;(x) II ~ 1.
(The proof
would now be complete if E was a strictly cconvex space). Now let! denote the algebra of all bounded linear mappings from Hoo(U) into itself. I..
is a Banach space and I', the identity
mapping from Hoo(U) into itself, is a real extreme point of the unit ball of
1:.
(see exercise 4.52).
and x £U let k(A)
=
Given 1/J£Hoo(U),II1/Jllu~ 1,
1/JCX+AW!;(X)) for all hD.
I t is
easily
seen that k£HCD;D). A further application of lemma 4.2 with z
= 21
yields
(* *) .
Now let L : HooCU) 1/Jo(I +
21
+
By (**)
w!;).
II I' + }CL  I') II
=
Hoo(U) be defined by the formula
sup 11/J(x) + }[1/JC x +}w!;cx))  1/J(x)] I ¢£Hoo(U) 111/J lIu
< 1,
X£ U
< 1.
Hence L
=
I'
and for any 6£E'
\\e have 0
1 = L(e)  6 = "2 eow!;.
By the HahnBanach theorem w!; is identically zero and as we have already noted this completes the proof. Other generalizations of Schwarz's lemma are also available and these together with the above have applications to Banach algebra theory.
In particular they yield a
generalised BanachStone theorem for J*algebras and a new proof of the RussoDye theorem. We now look at the maximum modulus theorem.
There are a
Chapter 4
164
number of different forms of the maximum modulus theorem discussed in the literature and here we confine ourselves to the following;
if f £ H(U), where U is a connected open
subset of ~, then either If(z) I has no maximum on U or f is a constant mapping.
We look at extensions of this result
by considering Banach valued holomorphic mappings defined on open subsets of [.
One can easily show that if f£ H(U;F),
U a connected open subset of [ Ilf(z)
II
11£11
has a maximum then
and f
a Banach space, and
is a constant.
Hence the
problem reduces to showing that the constant mappings are the only holomorphic mappings of constant modulus.
This is not
true in general, even for finite dimensional spaces, as the following example shows. Let f
Z;',
: D 
sup norm topology. but
Iif(z) II
=
Theorem 4.4
fez)
~ote
(l,z), where
Z;'
is ([2with the
that f is not a constant mapping
1 for all z E: D. Let E be a Banach space.
Each hoZomorphic
EvaZued mapping f defined on a connected open subset of ([ for u!hich
Ilf(z) II has a maximum 1:n constant i f and onZy i f E
is a strictly cconVex Banach space.
Proof
First suppose eE: E,
=
Ilell
extreme point of the unit ball. Ile+zx
II .::.
1 for all
z £II:,
some Izol .::. 1 we have
=
Choose x £ E such that
I z I .::. 1.
Ile+zox II
If
<
1 for
Ilell.::. t(lie+zoxll + Ilezoxll)
since this is impossible The function fez)
is not a complex
1,
e+zx,
lIe+zxll
=
1 for all z£ Il:,
<
1 and
I z I .::.
1.
is nonconstant but IIf(z) " is a constant function of z restricted to D and hence we Z £
D,
have proved the theorem in one direction. Now suppose E is a strictly cconvex Banach space. Let U be a connected open subset of 11:. suppose the mapping z E: U   IIf(z)
II
Let f£ H(U;E) and
is a constant mapping.
By using translations if necessary we may suppose DC:U and sup {lIf(z)II;z £D}.::. 1.
By lemma 4.2 and the HahnBanach
165
Holomorphic functions on Banach spaces
theorem Ilf(o)
+
A(f(z)f(o))ll~l
if IAI
~ 1;1~1,
D'.{o}.
Zs
Since the unit ball of E contains no complex extreme points it follows that fez)
f(o)
for all z near zero.
This
completes the proof. For arbitrary functions we also have the following method for recognising nonconstant holomorphic functions which are of constant modulus. Let E be a complex Banach space and let fez) non constant hoZomorphic mapping from D
= L~ a zn be a n n=o
[;Izi
{ZE
1
into E.
I}
<
Then a o Closed Span {aI' a 2 , . .} i f and only i f there exists an equivalent norm III ilion E and an open subset u of
D
IIlf(z) III is constant on
such that
U.
We now look at the CauchyHadamard formula.
This if (an)~=o
formula in one dimension states the following;
is a sequence of complex numbers and r = (lim sup I a I l/n)l nr
then the series h:o a n z for any ro
<
n
n
00
converges uniformly on {z;Izl ~ r } o
r.
The situation in infinitely many variables is quite different. Lemma 4.5 then
L'''
If E is a Banach space and el>n E E' for aU n el>n E H(E) i f and only i f el>n (x)    + 0 as n+ 00 n
n=o for every x in E Ci . e . el>n +
0
in the w* topology) .
oo
L
n eI> E H(E) then ,'" L. (eI> (x)) n z n converges n=l n n=l n for every x in E and z in [. By the CauchyHadamard formula
Proof
If
in one variable nlim _ sup
~
Conversely if el>n(x)
,00
n
then f = L. el>n n=l
E
lei>
+ 0
HGCE).
n (x)nll/n = lim
n~
as n +
ro
I eI>
n (x)
I
=
O.
for every x in E
Since the nth derivative of f at 0
Chapter 4
166
is ~n and this is continuous we may apply theorem 2.28 to n
complete the proof. Example 4.6 Let E be a separable Hilbert space with oo Let f(L z e) roo zn for orthonormal basis (e )00_ , n n 1 n=l n n n=l n all
L z e sE. n=l n n oo
\,00
Hence f = L n=l
at the nth coordinate of E.
n
~n
where
Since
~n
~n+
is evaluation 0
as n +
in (E',cr(E',E)) lemma 4.5 implies that fSH(E). · 1 an d 1 ~m n >
sUPl1 00
dAnf(o)
__ II l/n
00
However
1·
n!
Example 4.6 shows that in infinite dimensions we have to distinguish between the "radius of pointwise convergence" and the "radius of uniform convergence".
A further concept
is the radius of boundedness which enters in a natural way and plays an important role in later developments.
Let U
be an open subset of a locally convex space E and let B be a balanced closed subset of E.
We let I f E is
a normed linear space and B is the unit ball of E then dB(~'U)
E.
~
is the usual distance of
to the complement of U in
Now let F be a Banach space and let fsH(U;F).
B radius of boundedness of f at
~,
sup {IAI;As,,~+ABCU, "f"~+AB
rf(~,B),
<
{I AI ; As 4: , f,; + ABCU and
is defined as
oo}.
The B radius of uniform convergence of f at de fin e d ass up
The
~,
Rf(~,B),
the T a y lor s e r i e s
0
is
f fat
f,; converges to f uniformly on f,;+AB}. Proposition 4.7
Let U be an open subset of a locally convex
space E. let F be a Banach space and suppose f e: H (U; F) . E;
£
U.
B is a closed balanced subset of E and r fC ~,B)
rfCCB) = RfCCB)
> 0
n = inf{dBCCU), (lim f(Olil/n) l} n _ sup lIa n! B 00
If then
167
Ho[omorphic functions on Banach spaces
We first note that if E = U then dBCs,U) = + 00 and the above may reduce to ~ = = ~. This however says
Proof
00
that f is bounded and the Taylor series converges uniformly on s + AS for every A £( i f and only i f lim sup IlcinfCs) Il lln =0: n+ oo n! B If
0
lal
<
rfCs,B) then
<
(by the Cauchy inequalities). Hence lim sup n _
Since rfCs,B) rfCCB)
~
and
1
00
Ia I
~
dBCs,U) we have shown that
in£{ dBCs,U),
(lim sup IlcinfCO Ill/n)l}. n 
n!
00
B
The above also shows on taking lal < la'i < rfCs,B), that WCs+ax) 
~:o dn~fO (~") lIuB
<
~ L~
n=m+l +) 0
as m
+
and hence rfCs,B)
00
Conversely i f B = Clim sup IIdnfCq n+
then there exists C II I ~nfCO n! C1£)13B
_<
>
0
0
<
£
An we havel!d ~~O liB
<
~ for
<
such that
CCl£)n for all n
CNote that since rfCs,B) every n).
II~/n)_l
and
n!
00
>
0
Hence if s + (l£)BBCU
1
Chapter 4
168
<
00
Since £ was arbitrary it follows that An l/n 1 (lim sup lid nf(.,llIIB ) }. n~c.:'I
Now suppose y < Rf(s,B). 1 im
Ilf (s+x)
Then
dnf(O (x) II B + n! x£y
m> '"
0
as m
+
00.
for all n sufficiently large, say n
2,
Hence for any sup IfCs+x)1 x£ ~B
0
<
~
<
y,
>
n
o
.
we have
n=o
and ~ .::. rfCs.B).
Since ~ and
y
were arbitrary this implies
that Rf(s,B) .::. rf(s,B) and completes the proof. Corollarl 4.8
If E is a locally convex space.
F is a
Banach space and K is a compact balanced subset of E then
lim n'J
Wnf(O) n!
Q:l
Proof
II~/n=
0
for every f£HCE;F).
Since a holomorphic function is continuous it is
bounded on each compact subset of E and the result follows from proposition 4.7. If E is a finite dimensional space then rf(s,B)=dB(s,U) for any bounded subset B of E, any open subset U of E and any f £ H (U).
For this reason the concept of radius of
boundedness is not interesting in finite dimensions.
The remainder of this section is devoted to various properties of the radius of boundedness.
These results
were all motivated by topological considerations, which we discuss in the next section, but are also of independent interest.
We restrict ourselves to entire functions on a
169
Hoiomorphic functions on Banach spaces
Banach space E with closed unit ball B. write rf(O
in place of rf(E;,B).
In this case we
Note that rf(O
is an
isometric property of the Banach space E and will change if the Banach space is renormed even with an equivalent norm. If f is the function considered in example 4.6 then proposition 4.7 shows that rf(o)
=
1 (and hence rf(E;)
<
00
for all E; £ E) and so f is unbounded on every ball of radius 1+£, £
>
0,
centered at the origin.
This also shows that
the Taylor series expansion at zero converges at all points of E but does not converge uniformly on any ball of radius greater than 1 centered at zero. Our next result on the radius of boundedness says that every infinite dimensional Banach space supports an entire function with nontrivial of boundedness.
(i.e. not identically + (0) radius
This is a consequence of the following
deep result. Proposition 4.9
If E is an infinite dimensional Banach
space then there exists a sequence in E', II
=
1 for al l
nand
> 0
as n
(