View of Biduality in Spaces of Holomorphic Functions

BIDUALITY IN SPACES OF HOLOMORPHIC FUNCTIONS

P. GALINDO¹, M. MAESTRE²AND P. RUEDA

Abstract

This paper contains characterizations of the bidual space of some closed subspaces ofHb…U†, the space of holomorphic functions of bounded type defined on an open subsetUof a Banach space X, whereUis either a bounded balanced open set or the whole spaceX.

In recent years several authors have dealt with the matter of describing the bidual of some given spaces of polynomials or holomorphic functions.

See, for instance [1], [4], [7], [8], [13], [14] and [15]. Prieto (Theorem 12 of [13]) states thatH_b…X†is isomorphic to the bidual space ofH_wu…X†, the space of holomorphic functions of bounded type which are weakly uniformly con- tinuous on bounded subsets ofX, if and only if the space of continuousm- homogeneous polynomials P…^mX† is isomorphic to the bidual space of P_wu…^mX† ˆ P…^mX† \ H_wu…X†,8m2N. According to her we have the follow- ing situation:P…ⁿX† ˆ P_wu…ⁿX†for every n0 and since these families are Schauder decompositions of H_b…X† and H_wu…X† respectively, apparently both of these spaces coincide algebraically and, hence, topologically. How- ever, H…C† and H…†, where is the open unit ball of C, have the same Schauder decomposition…P…ⁿC††_n but they are not topologically isomorphic (see the remark after Corollary 10.6.12 of [9] or Remark 5). This example shows that given two spaces with Schauder decomposition, to have a topo- logical isomorphism between them we have to assume a stronger condition than mere isomorphisms between the spaces forming the decomposition. In order to clarify this situation we isolate a subclass of Schauder decomposi- tions: the R-Schauder decompositions. Concerning that, our main result is Theorem 9.

Given a Banach space X and its dual space X, we denote by Pw…^mX† (Hw…X† respectively) the space of all continuous m-homogeneous poly-

1 Partially supported by DGICYT(Spain) pr. 91-0326.

2 Partially supported by DGICYT(Spain) pr. 91-0326 and pr. 91-0538.

Received November 11, 1996; in revised form June 4, 1997.

nomials onX (of all entire functions onX respectively) which areweak- uniformly continuous on bounded subsets ofX.

Valdivia in [14] proves that the bidual space ofPw…^mX†is isometric to its 0-closure inP…^mX†assuming thatPw…^mX†does not contain copies of`<sup>1</sup>, whereas in [15], he proves the analogue result withHw…X†. In particular, he obtains that if X is a Banach space such that X has the approximation property and H<sub>w</sub>…X† contains no copy of `¹, then the bidual space of H_w…X†is canonically identified withH_b…X†. By using his results for poly- nomials and the decomposition we introduce, we obtain a new proof of these results and extend them to holomorphic mappings on the unit ball.

Most of the notation is standard. We refer to [10] and [6] for definitions and properties of locally convex spaces and holomorphic functions on in- finite dimensional spaces respectively. For an arbitrary locally convex space E, E denotes the strong dual of E. In the sequel we use the notation …X;k:k†for an arbitrary complex Banach space, Bfor the open unit ball of X and U for an arbitrary balanced open subset ofX. k:k means the dual norm in X. We denote by P…^mX† the space of all continuous m-homo- geneous polynomials on X, and by P_wu…^mX† the subspace of P…^mX† whose polynomials are weakly uniformly continuous on bounded subsets of X. Hb…U†denotes the space of all holomorphic functions of bounded type onU, that is, the space of all holomorphic functions onU which are bounded on allU-bounded sets. We recall that theU-bounded sets are in the caseUˆX the bounded subsets of X, whereas, in the case of an arbitrary open set U, they are the bounded subsets of U whose distance to the boundary ofU is greater than zero. If A is a U-bounded set, we put kf kA:ˆsup_x2Ajf…x†j, f 2 Hb…U†. Hb…U† will be endowed with the topology b defined by the seminormsk:k_A, whereAruns over allU-bounded sets. It is well known that …H_b…U†; _b†is a Fre¨chet space. Iff 2 H_b…U†,P₁

mˆ0P_m…f†denotes the Taylor series of f at the origin. Let Hwu…U† denote the subspace of Hb…U† of all holomorphic functions on U which are weakly uniformly continuous on all U-bounded sets. If G is an open subset of X, Hw…G† is the subspace of H_b…G† of all holomorphic functions on G which are weak-uniformly con- tinuous on allG-bounded sets.

Let us recall that a sequence of subspaces …F_n;k:k_n†_n of a locally convex space F is a Schauder decomposition of F if each x2F can be written in a unique way as P₁

nˆ0xn where xn2Fn and the projections um: P₁

nˆ0xn2F ! P_m

nˆ0xn2F are all continuous.

Given a sequence of Banach spaces …E_n;k:k_n†_n and 0<R 1 the Ko«the sequence space¹…A_R;…E_n†_n† (where A_Rˆ f…rⁿ†_n:0<r<Rg) is the Fre¨chet spacef…xn†_n2Q

nEn:pr……xn†_n† ˆP₁

nˆ0kxnk_nrⁿ<1 8r; 0<r<Rg, endowed with the topology given by the seminormsfprg0<r<R. WhenEnˆCfor every

nˆ0;1;. . ., this Ko«the sequence space is denoted by ¹…A_R†. By Cauchy- Hadamard's formula ¹…A_R;…E_n†_n† ˆ f…x_n†_n2Q

nE_n: lim sup_nkx_nk¹⁼ⁿ_{n R}¹g.

One can readily check that for 0<R<1; ¹…AR;…En†_n† ˆ¹₁……m†_m2N;…En†_n† a finite type power series space and for Rˆ 1, ¹…A1;…En†_n† ˆ ¹₁……m†_m2N;…En†_n†an infinite type power series space (see [9] p. 211). Under the canonical identification x2E_n?…0;. . .;0;x;0;. . .† 2¹…A_R;…E_n†_n†, one can check that…E_n†_n is a Schauder decomposition of¹…A_R;…E_n†_n†.

In our process to obtain results on biduals of some spaces of holomorphic functions we have isolated certain properties of their natural, i.e. polynomial, Schauder decomposition, which hold for their biduals as well. These spaces turned out to be power series spaces of the above type and only to emphasize the role played by such decompositions, we will labelR-Schauderto any de- composition of a Fre¨chet spaceE satisfying condition…R†stated in the next theorem, which gives a characterization of those power series spaces that will be useful in the infinite dimensional holomorphy.

Theorem 1. Given …En;k:k_n†_n; ¹…AR;…En†_n†is topologically isomorphic to any Fre¨chet space E that has…En†_n as a Schauder decomposition and satisfies the following condition: …R† for every sequence …xn†_n, xn2En, the series P₁

nˆ0x_n converges in E if and only iflim sup_nkx_nk¹⁼ⁿ_{n R}¹. Proof. Let :Eÿ!¹…A_R;…E_n†_n† defined by …P₁

nˆ0x_n† ˆ …x_n†_n for all xˆP₁

nˆ0x_n2E:

Since for any sequence of non negative real numbers …_n†_n and any 0<R 1the conditionlim sup_n¹⁼ⁿ_n 1=Ris equivalent toP₁

nˆ0rⁿn<1 8r:0<r<R, the map is an algebraic isomorphism. We prove that is also a topological isomorphism. Indeed, for every 0<r<R the set fx2E:p_r……x†† 1g ˆT₁

lˆ0fxˆP₁

mˆ0x_m2E:P_l

mˆ0r^mkx_mk_m1g is a barrel in the Fre¨chet space E; hence a neighbourhood of zero. Thus is continuous and therefore, as a consequence of the open mapping theorem, will be a topological isomorphism.

In Examples 2 and 4 below it is used that …P…^mX††_m(…P_wu…^mX††_m) is a Schauder decomposition of Hb…U† [13] (Hwu…U† [2], [13] respectively). If f 2 Hwu…X†, Aron proved in Proposition 1.5b of [2] thatPm…f† 2 Pwu…^mX†by using Cauchy's integral formula. It is well known that the same argument also works in the case of non entire functions and it can also be used to prove that…P_w…^mX††_m is a Schauder decomposition ofH_w…G†whenGis a balanced open set.

Examples 2. By using Cauchy inequalities we obtain that

a) The family…P…^mX†;k:kB†_mis an1-Schauder decomposition ofHb…X†.

b) The family…P…^mX†;k:kB†_mis anR-Schauder decomposition ofHb…RB†.

c) LetUX be a bounded balanced open set. The family…P…^mX†;k:k_U† is a 1-Schauder decomposition ofH_b…U†.

Remark 3. If…E_n;k:k_n†_n is anR-Schauder decomposition of the Fre¨chet spaceEandF is a closed subspace ofEso that…F\E_n;k:k_n†_nis a Schauder decomposition ofF, then…F\E_n;k:k_n†_n is anR-Schauder decomposition of F.

As a consequence of Remark 3 and Examples 2 we obtain:

Examples 4. a') …Pwu…^mX†;k:kB†_m is an 1-Schauder decomposition of H_wu…X†.

b')…P_wu…^mX†;k:k_B†_n is anR-Schauder decomposition ofH_wu…RB†.

c') Let UX be a bounded balanced open set. The family …Pwu…^mX†;k:kU†is a 1-Schauder decomposition ofHwu…U†.

a'') LetB be the open unit ball ofX. The family…Pw…^mX†;k:kB†_m is an1-Schauder decomposition ofHw…X†.

b'') Let B be the open unit ball of X and let R>0. The family …P_w…^mX†;k:k_B†_m is anR-Schauder decomposition ofH_w…RB†.

c'') Let G be a bounded balanced open subset of X. The family …Pw…^mX†;k:kG†_m is a 1-Schauder decomposition ofHw…G†.

Remark 5. Given 0<R<1 one can easily check that ¹…AR;…En†_n† is topologically isomorphic to ¹…A1;…En†_n†. Therefore a natural question arises: is it possible to establish a topological isomorphism between two Fre¨chet spaces having one an R-Schauder decomposition, 0<R<1, and the other one an 1-Schauder decomposition? Since we have defined R- Schauder decompositions having in mind the spaces of holomorphic func- tions of bounded type the question could be stated as: is it possible to find a Banach spaceX such thatHb…X†is topologically isomorphic toHb…B†? (We have pointed out above that this is not true whenX ˆC).

The answer to both questions is negative and has been given to us by Jose¨

Bonet in a personal communication which we gratefully acknowledge and where he pointed out the power series approach to the R-Schauder decom- positions. This is the way he proceeds:

A Fre¨chet space E is said to have property (DN) if given a fundamental system of seminorms…k:k_n†_n2Nthere is a continuous normk:konEsuch that for each n2N there exists m2N satisfying kxk_nskxk ‡s^ÿ1kxk_m8s>0 8x2E. Property (DN) is preserved under topological isomorphisms. The spaces ¹…AR†have not property (DN) when 0<R<1 and ¹…A1† has it (see Theorem 21.7.5 of [9]). Moreover it is not difficult to check that for any sequence of Banach spaces…E_n†_nand any 0<R 1the space ¹…A_R;…E_n†_n†

has property (DN) if and only if¹…A_R†has it. Thus ¹…A₁;…E_n†_n†is not to- pologically isomorphic to¹…A_R;…E_n†_n†for any 0<R<1. Therefore, given a Banach space X, the space Hb…X† …Hwu…X†;Hw…X†) is not topologically isomorphic toHb…B† …Hwu…B†;Hw…B†respectively).

Our next aim is to study when a topological isomorphism occurs between spaces E and F having R-Schauder decompositions of the same type. Al- though Theorem 8 below also follows from the power series spaces techni- ques, we confine ourselves to the context of R-Schauder decompositions since it is completely natural to the holomorphy and still keeps the paper self-contained without enlarging it too much.

Lemma 6. If…E_n†_n is an R-Schauder decomposition of E, 0<R 1, then …E_n†_nis anS-absolute decomposition of E.

Proof. SinceE is a Fre¨chet space, we just have to prove that …E_n†_n is an S-Schauder decomposition ofE (Proposition 3.10 of [6]).

Let…an†_n2 S:ˆ f…an†_nC: lim sup_njanj¹⁼ⁿ 1g and letxˆP₁

mˆ0xm2E.

We have to prove thatP₁

mˆ0anxn 2E. Since lim sup

n …ja_nj kx_n k_n†¹⁼ⁿlim sup

n ja_nj¹⁼ⁿlim sup

n kx_nk¹⁼ⁿ_n 1 R

it follows from the de¢nition of R-Schauder decomposition that P₁

mˆ0a_nx_n2E.

For 0<r<Rwe set B_r:ˆ fx2E:p_r…x† 1g: By Theorem 1, the family f¹_sB_rgs>0;0<r<Ris a fundamental system of neighbourhoods of zero. LetB_r be the polar of Br in E and let B_r be the polar of B_r in E. Since E is a Fre¨chet space, the family f¹_sB_r gs>0;0<r<R is a fundamental system of neigh- bourhoods of zero in it.

Lemma 7. i†Letm2E_m. Thenm2 B_r if and only ifkmk_mr^m. ii)IfˆP₁

nˆ0n2 B_r thenkmk_mr^m,8m2N.

Proof. i) Assume thatm2 B_r, then kmk_mˆ sup

xm2Em;kxmkm1jm…xm†j ˆ sup

xm2r^mBr\Em

jm…xm†j ˆr^m sup

xm2Br\Em

jm…xm†j r^m for allm2N.

Conversely, if m2E_m is such that kmk_mr^m, then given x2 Br, xˆP₁

nˆ0xn

j_m…x†j ˆ j_m…x_m†j 1

r^mk_mk_m1:

ii) this follows from (i) and the definitions ofB_randp_r.

Theorem 8. If …E_n;k:k_n†_n is an R-Schauder decomposition of E;

0<R 1, then…E_n ;k:k_n †_n is an R-Schauder decomposition of E. Proof. By Lemma 6, …En;k:kn†_n is an S-absolute decomposition of E.

Therefore …E_n;k:k_n †_n is an S-absolute decomposition ofE (Proposition 3.11 of [6]). Given 0<r<R, r<s<R and GˆP₁

mˆ0G_m2E, with G_m2E_m, there exists M>0 so that G2MB_s . By Lemma 7.i) _m2E_m satisfiesk_mk_m1 if and only ifs^m_m2 B_s\E_m. Hence,

r^mkGmk_m ˆr^m sup

kmk_m1jGm…m†j

ˆM r s

m sup

s^mm2B_s\E_m

G…s^m_m† M

M r s

msup

2B_s

G…† M

M r s

m:

Therefore the seriesP₁

mˆ0r^mkG_mk_m converges for all 0<r<R, and by the Cauchy-Hadamard formula,lim sup_m…kGmk_m†^1=m_R¹.

Now supposelim sup_m…kGmk_m†^1=m_R¹. We have to show that the series P₁

mˆ0G_m …E;E†-converges. Since the family fsB_rgs>0;0<r<R forms a fun- damental system of …E;E†-bounded sets, it is enough to prove that P₁

mˆ0G_m converges uniformly onsB_r, for alls>0 and 0<r<R. By Lem- ma 7.ii) if ˆP₁

mˆ0m2sB_r then kmk_msr^m, 8m2N. Hence jGm…†j ˆ jGm…m†j kGmk_mkmk_msrkGmk_m;8m2N: Thus P₁

mˆ0Gm

is uniformly Cauchy onsB_r, for 0<r<Rands>0. SinceE is a Fre¨chet space there existsG:ˆP₁

mˆ0G_m2E.

Theorem 9. Let…En;k:kn†_nand…Fn;k:kn†_nbe R-Schauder decompositions of the Fre¨chet spaces E and F respectively (0<R 1).

Assume that there exist algebraic isomorphisms T_m:E_mÿ!F_m 8m2Nso that:

i)In case 0<R<1, (Condition I)for each t>1there exist at;bt >0 such that

kTm…xm†k_matt^mkxmk_m and kxmk_mbtt^mkTm…xm†k_m 8xm2Em; 8m2N:

ii)In case Rˆ 1,(Condition II)there exist t;t⁰>0and a_t;b_t⁰ >0such that kT_m…x_m†k_ma_tt^mkx_mk_m and kx_mk_mb_t⁰…t⁰†^mkT_m…x_m†k_m 8x_m2E_m;8m2N:

Then the map T :xˆP₁

mˆ0xm2Eÿ!T…x†:ˆP₁

mˆ0Tm…xm† 2F is a to- pological isomorphism.

Conversely, if there exists a topological isomorphism T :Eÿ!F so that T…E_m† F_m, 8m2N, then T…E_m† ˆF_m and T_m:ˆTj_Em are topological iso- morphisms satisfying Condition I in case0<R<1and Condition II in case Rˆ 1.

Proof. To prove that T is well defined we have to show that P₁

mˆ0r^mkT_m…x_m† k_m converges for 0<r<R.

Suppose 0<R<1. Letr<s<R. By Condition I there exists a>0 so thatkTm…xm† kma…^s_r†^mkxmkm; m2N:Hence

r^mkT_m…x_m† k_m as^mkx_mk_m; m2N:

…1†

Now suppose Rˆ 1. Then by Condition II there exist s;a>0 so that kT_m…x_m† k_m as^mkx_mk_m; m2N:Hence

r^mkT_m…x_m† k_m a…sr†^mkx_mk_m; m2N:

…2†

Thus we obtain convergence in both cases.

Clearly, T is linear. By Theorem 1 the family fqrg0<r<R, where qr…y†:ˆP₁

mˆ0r^mkymkm; yˆP₁

mˆ0ym2F, is a fundamental system of continuous seminorms onF.

If 0<R<1; then it follows from (1) that T…fx2E:p_s…x† ¹_ag†

fy2F : qr…y† 1g: If Rˆ 1; then it follows from (2) that T…fx2E:prs…x† ¹_ag† fy2F:qr…y† 1g: These inclusions prove in both cases the continuity ofT.

LetVbe a map defined fromF intoEasV…P₁

mˆ0y_m† ˆP₁

mˆ0T_m^ÿ1…y_m†for every yˆP₁

mˆ0y_m2F. In an analogous way to that ofT it can be proved that V is well defined and continuous. One can easily check that V is the inverse map ofT.

Let us now show the converse statement. Define Tm:ˆTj_Em:Emÿ!Fm

for everym2N.T_mis a one-to-one linear mapping.

We now prove that each T_m is onto. Let y_m2F_mF. Since T is onto, there exists xˆP₁

nˆ0x_n2E such that T…x† ˆy_m. As T is continuous and linear ymˆT…x† ˆP₁

nˆ0T…xn† ˆP₁

nˆ0Tn…xn† and using the uniqueness of the above sum it follows that Tn…xn† ˆ0 for every n6ˆmand ymˆTm…xm† wherexm2Em. HenceTm…Em† ˆFm.

Finally we check that Conditions I and II are satisfied. Let 0<r<R.

SinceT is continuous there exists 0<s<R(we may suppose without loss of generality thats>r) and there exists a>0 such that T…fx2E:ps…x† ag†

fy2F :qr…y† 1g: Hence qr…T…x†† ¹_aps…x† 8x2E: In particular, if x_m2E_m; then q_r…T…x_m†† ¹_ap_s…x_m†, or equivalently, r^mkT_m…x_m† k_m

1as^mkx_mk_m. HencekT_m…x_m† k_m¹_a…^s_r†^mkx_mk_m:

i) Suppose 0<R<1. Ifrtends toR^ÿ;then stends to R^ÿ, hence ^s_rtends to 1^‡. Thus, givent>1 there existsat>0 so thatkTm…xm† kmatt^mkxmkm

8xm2Em:

ii) If Rˆ 1, then there exist t>0 and at>0 so that kTm…xm† km a_tt^mkx_mk_m 8x_m2E_m 8m2N.

Finally, sinceTis open there existb>0 and 0<s<R(we may choose with-

out loss of generalitys>r) so thatfy2F:q_s…y† bg T…fx2E:p_r…x† 1g†or equivalently,p_r…x† ¹_bq_s…T…x†† 8x2E:

Now in a similar way as above we get the two remaining inequalities.

Corollary 10. Let …E_n;k:k_n†_n and …F_n;k:k_n†_n be R-Schauder and R⁰- Schauder decompositions of E and F respectively (0<R;R⁰<1).

If there exists an algebraic isomorphism Tm:Emÿ!Fmfor every m2Nsuch that(Condition I')for each t>1there exist at;bt>0so that

kTm…xm† kmatt^m R R⁰

m

kxmkm and kxmkmbtt^m R⁰ R

m

kTm…xm† km; 8xm2Em;8m2N; then the map xˆP₁

mˆ0xm2Eÿ!T…x†:ˆP₁

mˆ0Tm…xm† 2F is a topological isomorphism.

Conversely, if there exists a topological isomorphism T:Eÿ!F such that T_m…E_m† F_m 8m2N, then T_m…E_m† ˆF_m and T_m:ˆTj_Em are topological isomorphisms satisfying Condition I'.

Proof. It follows from Theorem 9.

Corollary 11. Let …En;k:kn†_n and …Fn;k:kn†_n be R-Schauder decom- positions of E and F respectively (0<R 1). If E_n is isometrically iso- morphic to F_nfor every n2N, then E and F are topologically isomorphic.

In the sequel we apply the above results to our motivating spaces.

Let F be a closed subspace of …H_b…U†; _b†. The map given by x2Uÿ!_x2 F, where_x…f† ˆf…x†for allf 2 F, is a holomorphic mapping of bounded type [12] and its adjoint map is:F2 Fÿ!F2 Hb…U†.

Definition 12. The space F is said to be canonically isomorphic to a closed subspaceG of …Hb…U†; b† if :Fÿ!G is a topological isomorph- ism.

IfF ˆ Hb…U†, then the map is defined between Hb…U† andHb…U†. If we consider the spaceHb…U† as a subspace of its bidual space by means of the natural injection, the map is a projection. Hence, if is also one-to- one, the spaceH_b…U†is reflexive and is the identity map. In general, ifF is a closed subspace of …H_b…U†; _b†, such that is one-to-one and …F† ˆ F, then the spaceF is reflexive.

We denote _m:ˆj_Fm where Fm:ˆ F \ P…^mX†: Actually, _m is the ad- joint map of them-homogeneous polynomialm:x2Xÿ!m;x2 F_m, where _m;x…P_m† ˆP_m…x†, so _m:F_mÿ!P…^mX†: Also k_mk1 8m2N. If F_mˆ P…^mX†,_m:P…^mX†ÿ!P…^mX†is also a projection. Consequently, if_m is one-to-one thenP…^mX†is reflexive and _m is the identity.

Theorem 13. Let X be a Banach space and let U be either a bounded ba-

lanced open subset of X or UˆX. Let E and F be closed subspaces of …H_b…U†; _b†. PutF_m:ˆ F \ P…^mX†andE_m:ˆ E \ P…^mX†endowed withk:k_U if U is a bounded balanced open subset of X and withk:kBif U ˆX. Assume that…Em†_mand…Fm†_mare Schauder decompositions ofEandF respectively. If Em andFmare (canonically) isometrically isomorphic,8m2N, thenEandF are (canonically) topologically isomorphic.

Proof. It follows from Example 2, Remark 3 and Corollary 11.

Theorem 14. Let X be a Banach space, B the open unit ball of X and U either a bounded balanced open subset of X or U ˆX. LetF be a closed sub- space of…H_b…U†; _b†. Letbe a locally convex topology onH_b…U†weaker than or equal to _b. Let F_m:ˆ …F \ P…^mX†;k:k_B† if UˆX and F_m:ˆ …F\

P…^mX†;k:k_U†if U is a bounded balanced open subset of X. Assume 1)…Fm†_m2Nis a Schauder decomposition ofF;and

2) If f 2 F then Pm…f† 2 Fm for every m2N (equivalently, …Fm†_m is a Schauder decomposition ofF).

If there exist topological (canonical) isomorphisms T_m:F_mÿ!F_m sa- tisfying either Condition I if U is a bounded balanced open subset of X, or Condition II if U ˆX, 8m2N, then F is topologically (canonically) iso- morphic toF.

Conversely, if there exists a topological isomorphism T :Fÿ!F so that T…F_m† F_m,8m2N, then T…F_m† ˆ F_m and T_m:ˆTj_Fm are topological isomorphisms satisfying Condition I if U is a bounded balanced open subset of X, or Condition II if U ˆX.

Proof. LetRbe either1ifUˆX or 1 in other case. By Examples 2.a), 2.c) and Remark 3 the families …F_m†_m and …F_m†_m are R-Schauder decom- positions of F andF respectively. By Theorem 8, the family …F_m†_m is an R-Schauder decomposition ofF. An application of Theorem 9 completes the proof.

Corollary 15. Under the hypothesis of Theorem 14, if each F_m is iso- metrically (canonically) isomorphic to F_m, thenF is topologically (cano- nically) isomorphic toF.

We now see some applications of these results to the study of biduality of spaces of holomorphic functions.

Corollary 16. Let X be a Banach space. Let GX be either the open unit ball of X or GˆX. IfP_w…^mX†contains no copy of`¹for every m2N, thenH_w…G† is canonically isomorphic toH_w…G†⁰, the closure ofH_w…G†in …Hb…G†; 0†.

In particular, the isomorphism holds whenever X is an Asplund space.

Proof. To show that the conditions of Corollary 15 are fulfilled when F ˆ H_w…G† and ˆ₀, we need to check that P_mf 2 P_w…^mX†⁰ whenever f 2 Hw…G†⁰. If f 2 Hw…G†⁰ there exists a net …fi†_i2I in Hw…G† which 0- converges to f. By the Cauchy inequalities, for each m2N, the net …Pmfi†_i2I Pw…^mX†0-converges toPmf. HencePmf 2 Pw…^mX†⁰.

Since P_w…^mX† contains no copy of `¹, P_w…^mX† and P_w…^mX†⁰ are canonically isometrically isomorphic for every m2N (Theorem 2 of [14]).

Corollary 15 now implies that H_w…G† is canonically isomorphic to Hw…G†⁰.

IfX is Asplund, Pw…^mX†is Asplund too (Corollary 1.1 of [14], see also the proof of Theorem 5 in [4]),8m2N. Hence,P_w…^mX†contains no copy of`¹,8m2N. The first part of the Corollary now completes the proof.

Corollary 17. Let X be a Banach space such that X has the approx- imation property. Let GX be either the open unit ball of X or GˆX. If P_w…^mX†contains no copy of`¹for every m2N, thenH_w…G†is canonically isomorphic toHb…G†.

In particular, the isomorphism holds whenever X is an Asplund space such that X has the approximation property.

Proof. SinceX has the approximation property,Pw…^mX†andP…^mX† are canonically isometrically isomorphic (Theorem 3 of [14]). Examples 2.c), 4.c''), Theorem 8 and an application of Corollary 11 yield the result.

IfX is Asplund,P_w…^mX†is Asplund too (Corollary 1.1 of [14]) for every m2N. Hence, for every m2N Pw…^mX† contains no copy of `¹. The first part of the Corollary completes the proof.

Corollaries 16 and 17 have been obtained by Valdivia in [15] for entire functions under the assumption that`<sup>1</sup>is not contained in the space of entire functions. J.C. Diaz pointed out to us that this assumption is equivalent to the non-containement of`¹ inP_w…^mX† 8m2N(Corollary 1.25 of [11]).

Remark 18. Corollaries 16 and 17 hold for any open set G in X such that there exists a bounded balanced open subsetV of X satisfying that G coincides with the interior of V for the norm topology in X. Indeed, let k:k_V be the Minkowski gauge ofV inX. The Banach spaceY :ˆ …X;k:k_V†is topologically isomorphic to…X;k:k† and Gis now the open unit ball ofY. Moreover, Pw…† is isometrically equal to …Pw…^mX†;k:k_G† which clearly contains no copy of `¹. The conclusion follows from Corollary 16 (respec- tively Corollary 17).

Corollary 19. Let X be a Banach space and let U X be either a boun-

ded balanced open subset of X or UˆX. Assume that P_wu…^mX†contains no copy of`¹for every m2N(for example when X is an Asplund space). Then

a)Hwu…U† is topologically isomorphic toHw…U†⁰, where U is the in- terior on Xfor the norm topology of the closure of U for theweak-topology on X.

In particular, H_wu…B† is topologically isomorphic to H_w…B†⁰, where B and Bare the open unit balls of X and Xrespectively, andH_wu…X†is to- pologically isomorphic toH_w…X†⁰.

b)Moreover, if Xhas the approximation property thenHwu…U†is topo- logically isomorphic toHb…U†.

Proof. a) Since Pwu…^mX† and Pw…^mX† are isometrically isomorphic, Pw…^mX†contains no copy of`¹,m2N. Since the norm closureUofU agrees with theweak-closure ofU, by the bipolar theoremUis the bipolar of U in X. Thus, by Corollary 16 and Remark 18, H_w…U† and H_w…U†⁰ are topologically isomorphic. Now, sinceH_wu…U†and H_w…U† are topologically isomorphic, we finally obtain thatHwu…U†andHw…U†⁰ are topologically isomorphic.

b) An analogous proof to the one in Corollary 17 gives the conclusion.

Corollary 20. Let X be a Banach space and let U X be either the open unit ball of X or UˆX. If for every m2NP_wu…^mX† is (canonically) iso- metrically isomorphic toP…^mX†, thenH_wu…U†is (canonically) isomorphic to H_b…U†.

Proof. LetRbe either1if UˆX or 1 in other case. By Examples 2.a) and 2.c) …P…^mX††_m is an R-Schauder decomposition of H_b…U†. On the other hand, by Examples 4.a), 4.c) and Theorem 8,…P_wu…^mX††_mis anR-Schauder decomposition ofHwu…U†. Hence, Corollary 11 yields the result.

Corollary 20 and Corollary 21 below clarify Theorem 9 of [13].

Let us now consider the map e_m:z2Xÿ!e_m;z2 P…^mX†, where e_m;z…P† ˆeP…z† and where eP denotes the Aron-Berner extension [3] of P to X. Gonza¨lez in [7] has defined, extending an earlier definition of Aron and Dineen [4], a Banach space X to be Q-reflexive if the adjoint map e_m:P…^mX†ÿ!P…^mX† of e_m is bijective and hence, a topological iso- morphism for everym2N. Sinceke_mk 1, in order to satisfy the converse inequalities in the hypothesis of Theorem 9 one has to assumee_m to have some additional properties, for example to be isometries (in this case let us callX to be isometrically Q-reflexive). So as a consequence of Corollary 11 we get the following result (compare with Proposition 16 of [4]).

Corollary 21. Let X be an isometrically Q-reflexive Banach space and

let UX be either the open unit ball of X or U ˆX. Then the spaceH_b…U†

is topologically isomorphic toH_b…U†.

In fact, we can state the following theorem:

Theorem 22. Let X be a Banach space and let U X be either the open unit ball of X or U ˆX. Then the spaceH_b…U†is topologically isomorphic to H_b…U†if, and only if, X is Q-reflexive and the sequence…e_m†_msatisfies either Condition I if U6ˆX or Condition II if UˆX:

Acknowledgement. Thanks are given to Domingo Garc|¨a for the help- ful discussions we had with him during the preparation of this paper.

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