Some Topological Properties of Preduals of Spaces of Holomorphic Functions

Some Topological Properties of Preduals of Spaces of Holomorphic FunctionsAuthor(s): Christopher BoydSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 94A, No. 2 (Dec., 1994), pp. 167-178Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489483 .

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SOME TOPOLOGICAL PROPERTIES OF PREDUALS OF SPACES OF HOLOMORPHIC FUNCTIONS

By CHRISTOPHER BOYD

Department of Mathematics, University College, Dublin*

(Communicated by S. Dineen, M.R.I.A.)

[Received 20 January 1993 Read 30 November 1993 Published 30 December 1994]

ABSTRACT

For U balanced open in a Frechet space E, (2(U), T8) will denote the space of holomorphic functions on U equinped with the a-topology, while G(U) will denote the topological predual of (2(U ). We will show that (2(U), T.) satisfies the strict Mackey convergence condition if and only if E is quasinormable. Using this result, we show that G(U) is quasinormable if and only if E is quasinormable, and is Schwartz if and only if E is Schwartz. We then investigate necessary and sufficient conditions for G(U) to be nuclear and to have the approximation property.

Introduction

Let U be an open subset of a locally convex space E over C and let /(U) be

the space of holomorphic functions from U into C. We denote by To the compact

open topology on 2(U). A semi-norm p on 2(U) is said to be ported by the

compact subset K of U if, for each neighbourhood V, K c V C U, there is

CQ > O such that

p(f) ? CVIlf liv

for all f E 2(U). The i-z-topology on 2(U) is the topology generated by all semi-norms ported by compact subsets of U.

If K is a compact subset of E we denote by 2(K) the space of holomorphic

germs on K. The T,-topology on 2(K) is defined by

(AK), T,) = lim (2(U), T).

KC U

Let U be an open subset of a locally convex space E. We say that a semi-norm

p on E is in-continuous, if for each countable increasing open cover {U,}n of U

there is an integer no and C > 0 such that

p(f ) < Cllf IlUn

The subvention granted by University College, Dublin, towards the cost of publication of papers by

members of its staff is gratefully acknowledged by the Royal Irish Academy

* Current address Department of Mathematical Sciences, Dublin City University, Glasnevin,

Dublin 9

Proc. R. Ir. Acad. Vol. 94A, No. 2, 167-78 (1994)



168 Proceedings of the Royal Insh Academy

for every f in *Z(U). The n-topology on Z(U) is the topology generated by all T8-continuous semi-norms.

It is well known that an infrabarrelled locally convex space E is quasinormable

if and only if Eg satisfies the strict Mackey convergence condition. Aviles and Mujica [2] obtained a generalisation of this result where they showed that, if K is a

compact subset of a quasinormable Fr6chet space E, then (Y(K), r') satisfies the

strict Mackey convergence condition. A recent result of Bonet [5] shows that the

quasinormabilhty of E is also a necessary condition for r(K) to satisfy the strict

Mackey convergence condition. In this paper we show that, if U is a balanced open

subset of a Frechet space E, then (*(U), T.) satisfies the strict Mackey conver

gence if and only if E is quasinormable. To prove this result we give a new

description of the locally bounded sets of holomorphic functions on a balanced subset. This description may prove useful in other circumstances.

Mujica and Nachbin [14] have shown that, if U is an open subset of a locally

convex space E, then there is a complete locally convex space G(U) such that

G(U)'I = (r(U), T8). Specifically, G(U) is the space of linear forms on 2(U) whose

restriction to each locally bounded set is T0-continuous. In [6] the author shows that for each locally convex space E and each integer ni there is a complete locally

convex space Q(OE) such that Q(n EY, = (P('E), r.), and for each compact subset K of E there is a complete locally convex space G(K) such that G(K)Y1 = W(K),

T,,). The space Q(OE) is topologically isomorphic to the space of n-fold symmetric tensor products on E. If U (resp. K) is balanced open (resp. compact), then

{Q(nE)}) is an S-absolute decomposition G(U) (resp. G(K)) (see [6, props 4 and 5]). For U balanced open in a Frechet space we show that G(U) is quasinormable

if and only if E is quasinormable, and that G(U) is Schwartz if and only if E is Schwartz. We also investigate necessary and sufficient conditions for G(U) to be nuclear and to have the approximation property. We show that G(E) is nuclear if

and only if E is nuclear, and, when , = T, on r(U), we show that G(U) has the

approximation property if and only if E has the approximation property. We refer the reader to [8] for further reading on infinite-dimensional holomor

phy.

Spaces of holomorphic functions with the strict Mackey convergence condition

Let U be a balanced open subset of a locally convex space. We say that A c

4U) is locally bounded if for each x E U there is a neighbourhood VW of x and

Mx > 0 such that

lif Itv ?Mx

for all f E A. The following theorem gives another characterisation of locally bounded sets.



BOYD-Preduals of spaces of holomorphlc functions 169

Theorem 1. Let U be a balanced open subset of a locally convex space E. Then sets of

the form

(f:E dmf(O) vm?a}

where a > 0 and f V& }m is an increasing countable open cover of U, with each Vm a

balanced neighbourhood of 0, are a fundamental system of locally bounded subsets of 7r(U).

PROOF. We first show that every set of the form

(f:E dm f(O) v?a} (t 0 V.m iZ-g

is locally bounded. Let A be such a set. As {Vm}m is increasing, for x e U there is

an n e N such that x e V1. Then

djm f(0) Jm f (0) <a, m=n m! Vn m=n Vm

for every f e A.

For m = 0,..., n-1 there exists Am such that x S Am Vm. Then for f e A

am f(0) < Am a. m! Am V.m

Let Ux = AO VO n A1 V1 n ... AnI in-> n Vn. Then Ux is an open neighbour

hood of x, and

IIfll ? dm f(O) < a (? Am +1)

lfor every f s A, which proves that A is locally bounded.

Now suppose that B is a locally bounded subset of z(U). Let { ,ajOn denote the

sequence defined by AOt = = 1/2 and An = 1/n2"n for n ? 2. Then { ,ccA is an

increasing sequence which converges to 1 and En=0 An < ?

For every x e U we can find Ax > 1, a convex balanced open neighbourhood

U, of {Ax: I Al ? 1} in U and Mx > 0 such that

IIf Ikxu, ?MX



170 Proceedings of the Royal Irsh Academy

for all f E B. By the Cauchy inequalities we have

Jd1, f(0) <Ml/ m ! U. AX

for every integer m and f E B. As Mfr'rn I as m m and as Th Iwe can

find mo such that

dm f(0) i M! u.

Am

for all m > m. and f e B. For each n let

Vn = U{Ux | df(Q) <- m for all m ? n and fe B} U

2 i o

where U0 is a balanced neighbourhood of 0, such that

sup lIf luo < MO f EB

for some M0 < oo. Then {JV,}) is an increasing cover of U, consisting of balanced

neighbourhoods of 0. If f E B then

dm f(O) ||IAkIo)m M) M! V ax,M 120/ -inI

and therefore

m _ ? + E l+Eym <0 2 mU

We note that if E is a Banach space we can also assume that each ?,,, is

bounded. Let U be a balanced open subset of a locally convex space E. Then, since

((U), O) is bornological, we have that (Y(U), )O rlimn , (U)5 where the

inductive limit is taken over all absolutely convex r8-bounded subsets B of r(U). By [10, prop. 1] the r8-bounded subsets of Y(U) are locally bounded if and only if

this inductive limit is regular. When this happens the inductive limit need only be

taken over sets of the form

{f tE (U):E Jdn (o) ? <

where a > 0 and {IVm, is an increasing cover of U consisting of balanced



BOYD-Preduals of spaces of holomorphic functions 171

neighbourhoods of 0. Thus, when the in-bounded sets are locally bounded, we can write (f(U), 78) as the inductive limit lim B (U),, where we specifically know the

norm on each 4r(U)B B If E is a Frechet space with (Urn),,, a fundamental system of neighbourhoods of

0, then for every integer n we have (P(BE), T7) = lim m (P'E)1,, where

Bm n P E6 P(nE): {jPjtun ? 11.

If E is quasinormable, then Q(n E) = 0s,n, r E is also quasinormable for every

integer n and hence (P(nE), r,) satisfies the strict Mackey convergence condition. In fact one can say more, as the following theorem of Aviles and Mujica shows.

Theorem 2 [2]. Let E be a quasinormable Fr6chet space with fundamental system of

tneghbourhoods {Urn }n Then, for every p, there is a q such that, for every n, p( E)Bnf

and 'r. induce the same topology on the bounded subsets of P(CE)4q

Aviles and Mujica [2] use this result to show that, if K is a compact subset of

(luasinormable Frechet space, then 2'(K) satisfies the strict Mackey convergence condition. In [5], Bonet shows that the converse of this result is also true. For

balanced open sets we have the following analogous result.

Theorem 3. Let U be a balanced open subset of a Fr6chet space E. Then E is

quasinormable if and only if (r(U), T) satisfies the stnct Mackey convergence

condition.

PROOF. Let us first suppose that E is quasinormable and let A be a bounded subset of (t(U), in). Then A is locally bounded and, by Theorem 1, we may

aissume without loss of generality that

A (f O dm ffO) a

where a > 0 and ($,m}rn is a countable increasing balanced open cover of U with

each Vn a neighbourhood of 0. Since the VJ are increasing, we have

dm dn f) f(0)

fcEA m! VI

for every m e N. By Theorem 2 we can find a neighbourhood W of 0 such that

11 "lw and s, induce the same topology on every 11 tlv,-bounded subset of P(E) for all n. Let Wn =tam ,Vrn n (m + 1)W where (tUm)rn is as defined in Theorem 1.

Tohen (W.n}rn is an increasing cover of U and each W.. is a balanced neighbourhood of 0. To see that (W,n},, is a cover of U, we note that, given x e U, there is an m, sUch that wutI x e U for all m > ino. As {},, is an open cover of U, there is an

nt, such that tW' x E V,, for n > no. Finally, since {x} is bounded, there is an to such that x E (1, + 1)W. Setting j = max(nO, 1d we see that x E JI.




Let {f,} be a net in A which converges to 0 in 'ra and choose E > 0. We

choose m0 e N such that

d0$f(0) ? E ax$2

sup E m! g4m V.<

a Am < E

f c=-A M=MO =n

By [8, lemma 3.28] we have that

djm f(o) ->

0 m!

inr1 as ca for every m E- NJ. This implies that

dm f,(0) M ! 1w

as a --> oc for every m E N. Therefore, for every m, we can find am such that

dm fa(0) e |m i m+ m1)W 2m + 2

for a ? am. Choose a greater than any element of the set {a0o,. am 1}. Then

E m! w

m=QWm m< 2m +2 2

proving that fo --- 0 in (U)B, where

B =f:E dm f(0) <

Conversely, let us now suppose that (ff(U), T6) satisfies the strict Mackey convergence condition. Then E, =(P(E), -r) satisfies the strict Mackey conver

gence condition. By [3, appendix, prop. 3.7] it follows that E13 is boundedly

retractive and hence is sequentially boundedly retractive. Applying [5, theorem], we conclude that E is quasinormable. U

In proving Theorem 3 we also proved the following result.

Corollary 4. Let U be a balanced open subset of a Fr6chet space E. Then the inductive hmit (f(U), T) = lim > k'(U)B is boundedly retractwve if and only if E is quasi normable.

We also have the following corollary.



BoYD-Preduals of spaces of holomorphic functions 173

Corollary 5. If U is a balanced open subset of a Frdchet space E and r IS any topology on 2(U) such that r < r <Tm then (W(U), T) satisfies the stnct Mackey convergence condition if and only if E is quasinonnable.

PROOF. This result follows immediately from Theorem 3 and the fact that r. and

,r induce the same topology on the Tn-bounded sets of 2(U) (see [8, lemma 3.28]). U

We note in particular that the above result says that if U is a balanced open

subset of a Frechet space E, then %2(U), i-u) satisfies the strict Mackey conver

gence condition if and only if ? is quasinormable.

Quasinormable and Schwartz preduals of spaces of holomorphic functions

Using [6, prop. 11 and corollary 5] we obtain the following theorem, which

characterises when G(U) is quasinormable.

Theorem 6. Let U be a balanced open subset of a Fr6chet space E. Then E is

quasinormable if and only if G(U) is quasinormable.

PROOF. If G(U) is quasinormable, then Q(nE) is quasinormable for every integer n. In particular Q(E) = E is quasinormable. Conversely, suppose that E is

quasinormable. Then for every n, Q('E) is a quasinormable Frechet space.

Therefore, by [6, prop. 111, G(U); = (2(U), ), where r is a topology on 2(U)

finer than T, and weaker than r5. By Corollary 5, G(U)', satisfies the strict Mackey

convergence condition. Since by [14, theorem 4.4] G(U) is barrelled, it follows that G(U) is quasinormable. f

It is also possible to give a proof of Theorem 6 using the ideas in [9]; however,

the above proof is much shorter.

Theorem 7. Let U be a balanced open subset of a Fr6chet space E. Then G(U) is a

Schwartz space if and only if E is a Schwartz space.

PROOF. We first suppose that G(U) is a Schwartz space. Since by [14, prop. 2.6] E

is a subspace of G(U), it follows that E is also Schwartz.

Conversely, suppose that E is Schwartz. By Theorem 6, G(U) is quasinormable.

By [12, prop. 1.1.3.7.(2)], Q('E) is a Frechet Schwartz space for each integer n.

Hence, by [7, theorem 2], G(U) is also Montel and therefore Schwartz. U

To investigate when G(U) is nuclear we shall use the following theorem, which

will also be useful in obtaining results about the T6 topology.

Theorem 8. Let U be a balanced open subset of a Fr6chet space E. Then the following

are equivalent: (i) (2(U), T is dual-nuclear;

(ii) (Y(U), Ta) is dual-nuclear; (iii) G(U) is nuclear.



174 Proceedings of the Royal Irish Academy

PROOF. As G(U) is the completion of (V(U), ToY, we see by [15, props 5.1.1 and

5.3.1] that (i) and (iii) are equivalent. If (iii) holds, then G(U) is a complete barrelled nuclear space and hence is

reflexive. We therefore have G(U) = ((U), r8)Y,, by [7, theorem 3]. By [6, theorem

9], it now follows that (ii) holds. Finally, suppose that (ii) holds. Then (G(U)Y)b = (Z(U), T)b is a nuclear space.

By [7, lemma 1], G(U) is topologically isomorphic to a subspace of (G(U)Y)b and so is nuclear. U

We note that any of the above three conditions can only be satisfied if E is a

Frechet nuclear space. We now show that U = E, E nuclear, will imply all of the

above conditions.

Theorem 9. If E is a Fr6chet nuclear space, then G(E) is nuclear.

PROOF. By [17], E = N/F, where N is a Frechet nuclear space with an absolute

basis and F is a closed subspace of N. It follows, by [8, corollary 5.22], that

(Y(N), TO) is an A-nuclear space and therefore is a dual-nuclear space. By [16, theorem 11, (r(E), T,) is isomorphic to a closed subspace of (Z(N), T0), and therefore, by [15, prop. 5.1.2], is a dual-nuclear space. Using Theorem 8, we see

that G(E) is nuclear. U

We have a number of interesting corollaries.

Corollary 10. Let E be a Fr6chet nuclear space. Then (7(E), T.) is a dual-nuclear

space.

Corollary 11. If E is a 9Yi%; space, then X(OE) is a nuclear space.

PROOF. By [4, prop. 25], we have X(Od (7(E), r0Y,. The result now follows from Theorems 8 and 9. N

Preduals with the approximation property

The approximation property was introduced by Grothendieck in [12]. Every nuclear space has the approximation property. However, there are Frechet Schwartz which do not. In 1973, Enflo [11] gave an example of a separable Banach space which did not have the approximation property. In this section we will examine when G(U) has the approximation property for U balanced open in a Frechet space. We shall assume that r0 = T. on 7(U).

Proposition 12. Let E be a Montel locally convex space with 5-absolute decomposi tion {En },, and F any Banach space. Then {E,n cF},, is an 5-'absolute decomposition

for FeF.




PROOF. Since E is Montel, EeF is the space of all linear maps from E'

= E1 to F with the topology of uniform convergence on the equicontinuous (= bounded) subsets of E1. By [8, prop. 3.13], {(En)'b}n is an 9-absolute decomposition for E;. Let B = {xA}AE r be a bounded subset of E4. As in [8, prop. 3.13], B' = {n2afl x'jlA and B" = {n2 E_ xA}flA are bounded subsets of E1 for every (ant E 5'. If 4 E

EcF, let On = PI(E,)b. Then

||,0- E fkim =sup ( A < IIIIB" 0

as n -* c. Let Bn be the projection of B onto (E,,)b. If {0,a}a is a net in EcEF which converges to 0, then

11((A,)nI1Bn = SUp II4,a(Xn?)I < 2 11,a1IB '

A ncr,,

for all n. Hence (4')n 0 in En eF as a -O 0. This shows that (En eF} is a Schauder decomposition for EeF. Since

00 DC

E: la(nl| 110JIB =

F,SUp | ?(Aan XnA I n= n=lI A

x 1 1

< , - sup j4(n2a,xkA)J < II4IIB' E n=i A n

2

(En eF}n is an Y-absolute decomposition for E e F. U

Theorem 13. Let E be a complete Montel locally convex space with Y-absolute decomposition {En }n. Then E has the approximation property if and only if each En has the approximation property.

PROOF. If E has the approximation property, since each E,, is complemented in E, each E,, has the approximation property by [13, prop. 18.2.3].

Conversely, we now suppose that each En has the approximation property. Let F be a Banach space. Choose 4' E EEF, p a continuous semi-norm on EeF and S > 0. By Proposition 12 we may suppose that p satisfies

P(4) = E P(U,) n=O

for all 4 = E 4,,, E EeF. Choose m0 so that

00

E POO',,) < n =mO




Since each E, has the approximation property, En X F is dense in E, eF. There fore we can find %n E En 0& F such that

P Q qfn7-q) < n+2

Therefore

mO-1 mO-1 oo mO-i

P(m i E ? p( -) E p('J,) E 22n+2 2 < . \n-O / n=0 ln=m0 n=O

Since En (o F is a subspace of E ? F, En0 7 - E E 0 F. Therefore E X F is dense in E E F for every Banach space F. Applying [13, theorem 18.1.8] we see that E has the approximation property. U

Theorem 14. Let E be a Fr6chet Montel space such that G(E) is Montel; then the following are equivalent:

(a) E has the approximation property; (b) (ffU), '-) has the approximation property for one (and hence every) balanced

open subset U of E;

(c) (f(U), m) has the approximation property for one (and hence every) balanced open subset U of E;

(d) (X(K), r,) has the approximation property for one (and hence every) balanced compact subset K of E;

(e) G(K) has the approxumation property for one (and hence every) balanced compact subset K of E;

(f) G(U) has the approximation property for one (and hence every) balanced open subset U of E;

(g) E, has the approximation property; (h) (G(U), z0) has the approximation property for one (and hence every) balanced

open subset Uof E4;

(i) (r(K), m) has the approximation property for one (and hence every) balanced open subset K of E?;

(j) G(U) has the approximation property for one (and hence every) balanced open subset U of F;

(k) G(K) has the approximation property for one (and hence every) balanced compact subset K of E13.

PROOF. If E has the approximation property then each Q(QE) has the approxima tion property by [13, corollary 18.2.9 and prop. 18.2.3]. Therefore, by Theorem 13 and [7, theorem 2], G(U) and G(K) have the approximation property for every balanced open subset U and every balanced compact subset K of E. As E is a complemented subspace of both G(K) and G(U), this shows that (e) and (f) are both equivalent to (a). Since Q(CE) is a Frechet Montel space for each n it follows that (P(CE), s,) is Montel. If (a) holds, then it follows [13, corollary 18.1.7] that each (P(n E), ,,) has the approximation property. Therefore, by Theorem 13,




(7(K), T0) and (7(U), m) both have the approximation property. If U is balanced

open in E then

((U), TO)= lim (7(K), TO). Kc U

K balanced

Since the projective limit of spaces with the approximation property has the approximation property, (7(U), ;) will have the approximation property. As Eb is complemented in (W(U), O), (7(U), T8) and (7(K), ) we see that (b), (c) and

(d) each imply (g). Suppose that (g) holds. It follows by [13, corollary 18.2.9 and prop. 18.2.3] that

QC'E,) has the approximation property for each integer n. Since E1 is comple mented in G(U) and G(K) for U balanced open in Eb, and K balanced compact in Eb, it follows by Theorem 13 that (g), (j) and (k) are equivalent. For each integer

n, (PC'E), r,) = QCE;);. By [13, 15.6.8] QQ'E;) is a 9x.Y-# space and so it follows

from [13, 18.1.7] that (P(nE), T.) has the approximation property. As (7(U), ro) (resp. (7(K), T0)) are complete Montel spaces for U (resp. K) balanced open (resp. compact) in E1, by Theorem 13, (g) implies (h) and (i). Finally, since E is complemented in (7(U), r0) and (7(K), TO) for U (resp. K) balanced open (resp.

compact) in E;, (h) and (i) both imply (a). This completes the proof U

In particular, from Theorem 14 we have that (7(U), TO) and (7(U), T,) have

the approximation property for every U (resp. K) balanced open (resp. compact) in a Frechet Schwartz or 259# space with the approximation property.

The equivalence of (a) and (b) for any Frechet space was noted in remark 5

following [1, prop. 2.2].

ACKNOWLEDGEMENTS

The results in this paper were proved in my doctoral thesis submitted to the

National University of Ireland in August 1992. I should like to thank my supervisor, Professor S. Dineen, for his assistance and encouragement and Professor J. Mujica for his comments.

REFERENCES

[1] Aron, R. and Schottenloher, M. 1976 Compact holomorphic mappings on Banach spaces and the

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[2] Aviles, P. and Mujica, J. 1977 Holomorphic germs and homogeneous polynomials on quasi normable metrizable spaces. Rendiconti di Matem?tica 6 (10), 117-27.

[3] Bierstedt, K. D. 1988 An introduction to locally convex inductive limits. In H. Hogbe Nlend (ed.), Functional analysis and its applications, 35-133. Singapore/New Jersey/Hong Kong. World

Scientific Publ. Co.

[4] Boland, P. and Dineen, S. 1978 Holomorphic functions on fully nuclear spaces. Bulletin de la

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[5] Bonet, J. 1991 A question of Valdivia on quasinormable Fr?chet spaces. Canadian Mathematical

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178 Proceedings of the Royal Irnsh Academy

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Mathematik 41, 169-75



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Some Topological Properties of Preduals of Spaces of Holomorphic Functions

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