To what extent does the dual Banach space E t determine the polynomials over E?

(1)

Ark. Mat., 38 (2000), 343-354

To what extent does the dual Banach space E t determine the polynomials over E ?

Silvia Lassalle a n d Ignacio Z a l d u e n d o

A b s t r a c t . We show that under conditions of regularity, if E I is isomorphic to F I, then the spaces of homogeneous polynomials over E and F are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.

1. I n t r o d u c t i o n

In a recent p a p e r [DD] Dfaz a n d Dineen show t h a t if E ~ is isomorphic t o F ~, a n d E r has t h e Schur p r o p e r t y a n d the a p p r o x i m a t i o n property, t h e n for a n y n t h e spaces of n - h o m o g e n e o u s p o l y n o m i a l s over E a n d F are isomorphic. T h u s , in a sense, t h e dual spaces d e t e r m i n e t h e p o l y n o m i a l s over t h e spaces. T h e p u r p o s e of this n o t e is t o investigate f u r t h e r conditions assuring t h e existence of i s o m o r p h i s m s between spaces of polynomials. We also look into t h e p r e s e r v a t i o n or n o n - p r e s e r v a t i o n of some classes of p o l y n o m i a l s by these isomorphisms, since it has seemed t o us t h a t t h e mere fact t h a t two spaces of p o l y n o m i a l s are isomorphic does n o t do justice t o t h e rich s t r u c t u r e of such spaces (note for e x a m p l e t h a t p ( k R n ) is isomorphic to

p(n-lRk+l)).

In Section 2 we show t h a t we c a n assure t h e existence of a n i s o m o r p h i s m u n d e r regularity c o n d i t i o n s on t h e spaces. Recall t h a t a B a n a c h space E is called Arens-regular if all linear o p e r a t o r s E - - + E ~ a r e weakly c o m p a c t , a n d s y m m e t r i c a l l y A r e n s - r e g u l a r if this is so for all s y m m e t r i c linear o p e r a t o r s (see [AGGM] a n d [GI]).

I n Section 3 we investigate t h e preservation of some subspaces of p o l y n o m i a l s u n d e r t h e m a p s defined in Section 2. We f i n d - - w i t h no condition on E or F - - t h a t if E r is isomorphic t o F ~, t h e n t h e spaces of n - h o m o g e n e o u s integral polynomials over E a n d F are isomorphic; a n d t h e s a m e holds t r u e for t h e spaces of weakly c o n t i n u o u s polynomials. Finally, in Section 4 we give some examples.

(2)

344 Silvia Lassalle and Ignacio Zalduendo

T h e authors are grateful to S. Dineen for introducing them to the problem, and for helpful conversations.

Added in proof. We have learnt that F. Cabello S~nchez, J. M. F. Castillo and R. Garcfa have independently obtained T h e o r e m 4, see Theorem 1 in [CCG].

2. ~ a n d t h e A r o n - B e r n e r e x t e n s i o n

Before going into the problem, we discuss some preliminaries and fix notation.

Throughout, E and F will be real or complex Banach spaces. By JR we will denote the canonical inclusion of F in its bidual F " . By 7)(nE) we denote the space of all n- homogeneous continuous scalar-valued polynomials over E. This is easily seen to be isomorphic to the space of all continuous symmetric n-linear forms, which we denote by s (E). Thus each polynomial P has an associated symmetric multilinear form A such that P ( x ) - - A ( x , ..., x). T h e r e are many interesting subclasses of polynomials.

Among them: P ~ ( n E ) , the class of those polynomials which are weakly continuous on bounded sets; PI('~E), the class of integral polynomials; and 7~wsc(nE), the class of weakly sequentially continuous polynomials. For more on polynomials over Banach spaces, see [D], [M] and [GJL].

Our approach to the problem will be via the following construction. Any continuous linear morphism s: E ' - + F t induces a continuous linear map

in the following way. If y is an element of F , define the linear morphism

~): s - - ~ / : k - l ( E )

by y(B)(Xl, ^{. . .} , X k _ l ) m S ( B z l ... zk-1)(Y) , where Bxl ... x~_l is the element of E ' obtained by fixing the k - 1 variables Xl, ... ,X~-l. Now if P is an n-homogeneous polynomial over E, and A is its associated symmetric n-linear form, A can be assigned an n-linear form g(A) over F by setting

~(A)(yl,..., yn) -- (91 . . . 9n)(A).

Note t h a t ~(A) need not be symmetric. We can, however, define an n-homogeneous polynomial over F by putting $(P)(y)=~(A)(y, ..., y).

In the case S=JE,: E'--~E'" (the natural inclusion), the morphism $ obtained is the well-known Aron-Berner extension of polynomials from a Banach space E to its bidual ([AB], [Z2]). In this particular case we will use the notation P and

(3)

To w h a t e x t e n t d o e s t h e d u a l B a n a c h s p a c e E ~ d e t e r m i n e t h e p o l y n o m i a l s over E ? 345

for g(P) and g(A) respectively. T h e lack of s y m m e t r y of .4 is at the heart of the m a t t e r t h a t concerns us here, so we take a moment to refer to some properties of A.

First, although A need not be symmetric, the elements of E and those of E " can always be permuted in the variables of A. Also, A is always weak* continuous in its first variable. It can be seen ([~r], [ACG]) t h a t s y m m e t r y of A is equivalent to its weak* continuity in all variables, and also to the weak compactness of the operator E - + 7 ) ( ~ - I E ) associated to P. S y m m e t r y of A is always obtained if the space E has the property of symmetric regularity mentioned above.

Since the A r o n - B e r n e r extension is a well-studied object, we will find it very convenient to write g and g in terms of the A r o n - B e r n e r extension. This is what we do in the following lemma.

L e m m a 1. For any Yl, ..., Yn in F, and all symmetric n-linear forms A over E, g(A)(yl .... , yn)=-4(S'(JF(Yl)), ..., S~(JF(yn))). In particular, g ( P ) = P o s ' o J F .

Proof. We proceed by induction on n. If A C E ' , we have ~ ( A ) ( y ) = s ( A ) ( y ) = J F ( y ) ( s ( A ) ) = s ' ( J F ( y ) ) ( A ) = A ( s ' ( J F ( y ) ) ) . Now suppose t h a t the result is true for ( n - 1 ) - l i n e a r forms. We first show that the ( n - 1 ) - l i n e a r form over E " obtained from A by fixing s'(JF(yn)) in the last variable coincides with ~)~(Jl):

Let Zl, ... ,Zn-I E E ~. We have

A(z~, ..., zn-~, (s'(JF(yn))) = (z1 . . . ~ n - 1 ) ( ( s ' ( J F ( y n ) ) ( A ) ) - ) and

ftn(A)(zl, . . . , Z n - - 1 ) = (Z'I . . . 5 n - ~ ) ( ~ ) ~ ( A ) )

(where the tildes over elements of E " refer to J~,). Thus it will be enough to check t h a t the ( n - 1 ) - l i n e a r forms over E, ( s ' ( J F ( y , O ) ( A ) ) - and ~),(A), coincide. Let xl, ... , X n - I C E . T h e n

( s ' ( J F ( Y n ) ) ( A ) ) - ( X l , . . . , x n - 1 ) =S'(JF(yn))(A~: 1 ... ~._~) = JF(yn)(s(A~a ... ~._~))

: s ( A x l . . . , ) ( y , ) = y n ( A ) ( x l , . . . , Xn--1 ).

Now, using our inductive hypothesis

g(A)(yl, ..., Yn) -- (Yl . . . yn)(A) ^: (YI . . . yn-1)(~ln(A))

---- 8 ( Y n ( A ) ) ( Y l , ... , Y , - I ) : 9 , ( A ) ( s ' ( J F ( y l ) ) , . . . , S ' ( J F ( Y n - 1 ) ) )

= A ( s ' ( J F ( Y i ) ) , . . . , S ' ( J F ( Y n ) ) ) . []

In what follows we will usually write y instead of JF(Y), for elements y E F , even when we consider the elements of F ~ via the natural inclusion.

(4)

C o r o l l a r y 2. I r A is symmetric, then ~ ( A ) = A o ( s ' x . . . x s ' ) . Thus ~(A) is also symmetric, and if P is the homogeneous polynomial associated to A, then $ ( P ) = Po81 .

Proof. Clearly ~(A) is symmetric, for

~(A)(yl, ..., y~) = ~i(s'(yl), ..., s'(y~)).

Let Yl,..., Yn denote elements of F , and w l , . . . , wn elements of F " . We will show, by induction on k, t h a t

~(A)(w,, ... , wk, Y k + l , "" , Yn) = -~(S' ( W l ) , . . . , 8' ( W k ) , s' ( Y k + l ) , "", 8' ( y n ) ).

Recall t h a t the A r o n - B e r n e r extension of any s y m m e t r i c n-linear form is weak* continuous in its first variable, and furthermore elements of F and F " always commute.

Also, s' is weak*-weak* continuous. Consider a net (y~) of elements of F , weak*

converging to wl. Then, for k = 1,

~(A)(wl, y~, ..., Yn) = lim ~(A)(y,~, Y2, .--, Yn) = lim ~(A)(y~, y2, ..., yn)

o~ ot

= lim ]i(s'(y,), s'(y2), ..., s'(y,~)) = .4(s'(wl), s'(y2), ..., s'(yn)).

Now suppose the equality holds for k - 1 . We have

~(A)(wl, ..., wk, Yk+l,..., yn) = lim ~(A)(y~, w2, ..., wk, Yk+l,..., Yn)

= lim ~(A)(w2, ..., wk, y~, yk+l,..., Y~)

= l i m / ~ ( s ' ( w 2 ) , . . . , s ( k ) , ' w s' (y~), s' ( y k + l ) , . . - , s ' ( y ~ ) )

= lim A(s'(y~), s'(w2), . . , s'(w~), s'(y~+~), ^{. . ,} s'(y~))

= i i ( s ' ( ~ ) , s'(~2), ..., s'(~k), s'(yk+~) .... , s'(~n)).

Therefore ~(A)(wl, ... , w n ) = f i ( s ' ( w l ) , ..., s'(wn)), and $ ( P ) ( w ) = P ( s ' ( w ) ) . []

C o r o l l a r y 3. Let PET)(nE), and let A be its associated symmetric n-linear form, and suppose that s: E~-+ F ~ is an isomorphism. Then if ft is symmetric,

(s -1 o ~ ) ( p ) - - P .

Proof. Note t h a t ~(A) is symmetric. T h u s for Xl, ... , x n E E , we have s - l ( s ( A ) ) ( X l , ..., xn) = 8(A)((s-1)'(Xl), ..., (S--l)'(Xn))

= A ( $ ' ( ( s - 1 ) ' ( X l ) ) , . . . , 8 ' ( ( 8 - - 1 ) ' ( X n ) ) )

= A(x~,..., xn) = A ( x l , . . . , xn).

Since ~(A) is the s y m m e t r i c n-linear form associated to ~(P), for a n y x E E we have s - l ( ~ ( P ) ) ( x ) = P ( x ) . []

(5)

To w h a t e x t e n t d o e s t h e d u a l B a n a c h s p a c e E ~ d e t e r m i n e t h e p o l y n o m i a l s over E ? 347

T h e o r e m 4. If E and F are symmetrically Arens-regular, and E ~ and F ~ are isomorphic (resp. isometric), then for any n, 7m(nE) and P('*F) are isomorphic ( resp. isometric).

Proof. Let s:E~-+F ~ be the isomorphism. Since E is symmetrically Arens- regular, for a n y s y m m e t r i c n-linear form A over E we have t h a t A is symmetric.

Thus for any n-homogeneous polynomial P over E, ( s - l o $ ) ( P ) = P . Analogously, for any n-homogeneous polynomial Q over F , (~os-1)(Q)=Q. Note also t h a t

II~(P)l[ = IIP~176 ~-- ]]P]I Hsll n, and the same for s -1 and Q. []

Remark 1. If one has m o r p h i s m s s: E~--~F ~ and t: F~--~G ~, and A is the symmetric form associated to a homogeneous polynomial P , then s y m m e t r y of A implies (~o$) ( p ) = ~ - ~ ( p ) .

Remark 2. It is easy to see t h a t if E ~ and F ~ axe isomorphic, and one is Arens- regular, then so is the other. Indeed, say s: E~--+F ~ is an isomorphism, and F is Arens-regular. If T: E--~E ~ is a linear m a p , consider the d i a g r a m

T r

E " > E I

F " " F I

T/"

F

where L=soT'oS%JF. Since L is weakly compact, its bitranspose L ' , has range in F ' by G a n t m a c h e r ' s theorem. Thus T~=s-loL"o(s~) -1 is weakly compact, since L " is. T h e n so is T. Thus s y m m e t r i c regularity of b o t h spaces in the t h e o r e m can be replaced by regularity of one of them.

Remark 3. Note t h a t for S=JE,, Corollary 2 recovers the following result of [AGGM]: If E is symmetrically regular, t h e n P = P o o , where 0: EiV-+E" is the restriction map; thus there are no new 'evaluations' beyond E ' .

3. ~ a n d s o m e s u b s p a c e s o f p o l y n o m i a l s

Since s: E'--+F' is a m o r p h i s m between dual Banach spaces, it is n a t u r a l to expect t h a t those types of polynomials over E which are more closely related to E '

(6)

will be preserved by the morphism g:

P(nE)-+P(nF).

T h e formula

g(P)=Pos'oJF

proved in the previous section shows immediately t h a t finite type, nuclear, and approximable polynomials are all preserved by ~. In this section we show t h a t the same is true for weakly continuous and integral polynomials, and t h a t if

E' and F'

are isomorphic, then

7)~(nE)

is isomorphic to

P~o(~F),

and :Pt(nE) is isomorphic to

PI(nF),

with no further assumptions on E or F . Perhaps surprisingly, the class of weakly sequentially continuous polynomials is not, in general, preserved by g.

We provide examples of this situation in the last section.

Recall that weakly continuous polynomials over E can be characterized ([AG], IT]) as those for which there exists a compact set K contained in E ' for which

IP(x)l _< c sup I~(x)l"

~ c K

holds for all

x EE.

We will denote the seminorm on the right-hand side of the inequality by IlxllK, and say that P is K - b o u n d e d . T h e smallest possible c is called the K - n o r m of P and denoted

IIPIIK.

L e m m a 5.

If P is K-bounded, then $(P) is s(K)-bounded and

II~(P)II,(K) ~ IIPIIK.

Proof.

It was proved in [AG], [CDDL] t h a t P is K - b o u n d e d if P is and t h a t

IIPIIK=IIPIIK.

Thus for any

y e F ,

I~(P)(y)I =

I(P~176 <

IIPIIK sup

I~((s'~

"~,EK

= IIPIIK sup Is('y)(Y)I" = IIPIIKIlYlI,~K)" []

"~EK

Thus g preserves weakly continuous polynomials.

P r o p o s i t i o n 6.

If s: E'--~ F' is an isomorphism (resp. isometry), then

~:Pw("E)

~ P w C F )

is an isomorphism (resp. isometry).

(7)

To what extent does the dual Banach space E' determine the polynomials over E? 349

Proof.

Given

PEPw(nE),

its associated linear o p e r a t o r

E-+P(n-IE)

is compact ([AHV]). T h u s the A r o n - B e r n e r extension of its associated n-linear form is s y m m e t r i c a n d we have, by the results of t h e previous section,

(s-Io$)(P)=P.

Analogously, for Q e : P ~ ( n F ) , one has

(gos-1)(Q)=Q.

Note also t h a t II~(P)II =

IlPos'ogFII <_

Ilell IlsllL

and the same for s -1 and Q. []

Recall t h a t a polynomial

PCP('~E)

is called integral if there is a regular Borel measure # on

BE,

(the unit ball of E ' in its weak* topology) such t h a t for all

xCE,

P(x) = [ 7(x) nd#('y).

J B E l

T h e measure # is said to represent P, and the infimum of the total variations of the measures representing P is the integral norm of P , denoted IIPIIr. It was proved in [CZ] t h a t the A r o n - B e r n e r extension of an integral polynomial is integral. Indeed, if # is a measure representing P , and one defines

by

U: L 1(#) > E '

u(f)(x)

^{= 9}f /(7)~(x) dr(7),

d / 3 E l

t h e n U is a norm-one m a p and the A r o n - B e r n e r extension of P m a y be written

P(z) = IBm, U'(z)n d#.

W i t h this notation, we prove the following lemma.

L e m m a

7. If P is integral, then ~(P) is integral, and II~(P)ll,<lHlnllPII1.

Proof.

For y E F ,

= P(s'(JF(y))) =

U ' ( s ' ( J F ( U ) ) ) n

= fB , [(s~176

Thus, g ( P ) is integral, and

II~(P)lb _<

II(s~176 <_

Ilsll"l,I.

Since tiffs holds for any measure representing P,

II~(P)lb-< Ilsll"llPIIz. []

(8)

P r o p o s i t i o n 8. If s: E'--+ F' is an isomorphism (resp. isometry), then

~: 7~I("E) > P I ( n F ) is an isomorphism ( resp. isometry ) .

Proof. By L e m m a 7, $ and s -1 are b o t h morphisms between the spaces of integral polynomials. We show now t h a t $ ( P ) = P o g . Clearly P o g coincides with

$(P) when restricted to F . Thus it will be enough ([Z1}) to see t h a t the first-order differentials of P o s t have the properties

(a) for all y C F , D(Pos')(y) is weak* continuous;

(b) for all w E F " and (y~)CF, weak* converging to w, D(Pos')(w)(y~) --+

D(Pos')(w)(w).

But since ~Pos')(w)=fB~, U'(s'(w)) n d#, upon differentiating we have D(Pos')(w)(v) = n f U ' ( s ' ( w ) ) " - ' V ' ( s ' ( v ) ) d~,

J B s t

which is weak* continuous of the variable v.

Now

s - 1 ( s ( P ) ) = s ( P ) ~ 1 7 6 = P ~ 1 7 6 1 7 6 = P ~ = ^{= P .}

Analogously, for QE~Pz(nF), one has (~os-1)(Q)=Q. T h e norms of ~ and s - I are controlled by the inequality in L e m m a 7. []

Remark 4. Note t h a t the same type of result as in the propositions can be obtained for any subspace of polynomials as long as their associated linear operators E - + T ' ( n - ] E ) are weakly compact, and the A r o n - B e r n e r extension preserves the class.

In particular, we obtain the following results for extendible polynomials (see [ g a ] , [C] and [Z2] for the pertinent definitions).

L e m m a 9. If P is extendible, then ~(P) is extendible, and li~(P)lle_<

]]sii"llPll . []

P r o p o s i t i o n 10. If s: E~-~ F ~ is an isomorphism (resp. isometry), then

is an isomorphism (resp. isometry). []

(9)

To what extent does the dual Banach space E' determine the polynomials over E? 351 4. E x a m p l e s

In this section we give three examples. T h e first shows t h a t ~(P) m a y differ from P o s ' . T h e second t h a t ~ m a y differ from ~-o~, and the third t h a t even when

s: Xt--~Y I

is an isomorphism, ~ m a y not preserve the class of weakly sequentially continuous polynomials.

We mention a few facts and fix some notation before going into the examples.

If X is a B a n a c h space, consider the 2-homogeneous polynomial P defined over X x X ' by

P(x, xI)=xI(x).

It is easily seen t h a t P is weakly sequentially continuous if and only if X has the D u n f o r d - P e t t i s property. Also, one m a y check t h a t the A r o n - B e r n e r extension of P to X " x X "l is

~ z I ! I t l \ 1

tx , ~ ~ = ~ [ ~ " ' ( x " ) + x " ( e ( x ' " ) ) ] ,

where

Q:X"'-~X'

is the restriction (i.e., the transpose of the natural inclusion

Jx: X--rX").

In the first two examples we will use the notation

E = X x X I, F : X'I x X ', G = X " x X'"

and over these spaces we consider the polynomials

P(x, xl)~-x'(x), Q(x",x')-~x"(x I)

and morphisms

s: E ' ) F ' , t: F ' - ~ G ' given by

s=Jx,@idx,,,

t = i d x , , ,

@Jx,,.

Also, we let

r=J~(,.

Example 1. $(P)y~ Pos '.

We first calculate $(P). For (x",

x')EF,

we have

$(P)(x", x') = (Pos')(Jx,, (x"), Jx,(x')) = P(rJx,,(x"), Jx,(x'))

= P(x", Jx' (x')) = 1[Jx" (x')(l,) +x,i(9(jx,

(x,)))]

[ x " ( x ' + x 'I x'~l

= 5 ) ( - = q ( ~ " , ~ ' ) - Thus

~(P)=Q.

Now calculate $(P) a n d / 5 o s ' ,

~ ( p ) ( ~ , ~ , ~,i,) = 0 ( ~ , ~ , x'") = 1[~,~(~,,,)+~,,,(~(~,~))],

- - " 1 I f ? Z~J Z ~ I l l

(Pos')(xiV,x '') =P(r(x'V),x '') = ~[x ( r ( x ) ) + r ( x )(Q(x

))].

Thus

$(P)=Pos'

if and only if

x"(x'")=r(xi')(Q(x'")),

but this only holds for reflexive X .

Note t h a t we have seen t h a t

$(P)=Q.

If X has the D u n f o r d - P e t t i s p r o p e r t y but X ' does not, then P is weakly sequentially continuous, b u t Q is not. We also have the following result.

(10)

352 Silvia Lassalle and Ignacio Zalduendo

C o r o l l a r y 11. If X is an infinite-dimensional Banach space with the Dunford- Pettis property, then X • X ' contains 11 .

Proof. If E = X • ~ did not contain 11 then every weakly sequentially contin- uous polynomial would be weakly continuous [FGL}. Thus the operator E--rE' associated to the polynomial P above would be compact, its corresponding A would be symmetric, and g ( P ) = P o s ~ forcing X to be reflexive, and therefore finite- dimensional. []

Example 2. tos~tog.

First, calculate t'oJG,

(t% Jo)(x", x m) = t' ( gx,,(x"), gx,,,(xm) ) = ( Jx,,(x"), g~x,,( Jx,,,(xm) ) )

= ( J x " (x"), xm).

We have already calculated ~(P), so using this we have

Then

(tog) (P)(x", x ' ) = (g(P)ot'o J c ) ( x n, x " ) = g ( P ) ( J x " (x"), x m)

= 89 [gx', ( x " ) ( x " ) + x ' " ( r ( J x , , (x")))] = xm(x").

(t-~-s)(P)(x", x'") =

(Po(tos)'oJG)(x", x"') = (Pos')( (t'oJe)(x", x") )

= (Pos')(Jx.

(x"), x"') =

P(r(Jx-(x")), x"')

= p ( x " , x ' " ) =

thus tos=to~ if and only if x"(~(xm))=xm(x"), but again, this only happens for reflexive X.

Example 3. An isomorphism s: XP--~Y ~ such that g does not preserve the class of weakly sequentially continuous polynomials.

We begin with the well-known example of [S] of Banach spaces X and Y with isomorphic duals, such that X has the Dunford-Pettis property and Y does not:

X = ( ~ - ' ~ I ~ ) 1 and Y = X O I 2 . n>l

Call the isomorphism s: X'--+Y', and consider the 2-homogeneous polynomial Q over Y defined by Q(x, a)-~-~_>l a n. The operator - 2 Y - + Y ' associated to Q sends (x, a) to (0, a) and is therefore weakly compact. Then ( g o s - 1 ) ( Q ) = s o s - l ( Q ) = Q . Since X has the Dunford-Pettis property, all polynomials over X are weakly sequentially continuous [It], in particular (s-1)(Q) is. Thus g sends this weakly sequentially continuous polynomial onto Q, which is not weakly sequentially continuous (Q(0, en)--1 for all n).

(11)

To what extent does the dual Banach space E I determine the polynomials over E? 353 R e f e r e n c e s

[ A B ] A R O N , R. and BERNER~ P., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3-24.

IACG] ARON, R., COLE, B. and GAMELIN, T., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51-93.

l A G ] A R O N , R. and GALINDO, P., Weakly compact multilinear mappings, Proc. Ed- inburgh Math. Soc. 40 (1997), 181-192.

[AGGM] ARON, R., GALINDO, P., GARCfA, D. and MAESTRE, M., Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soe.

348 (1996), 543-559.

[AHV] ARON, R., HEaVeS, C. and VALDIVIA, U., Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189-204.

[CCG] CABELLO SANCHEZ, F., CASTILLO, J. i . F. and GAaCiA, R., Polynomials on dual-isomorphic spaces, Ark. Mat. 38 (2000), 37-44.

[CDDL] CARANDO, D., DIMANT, V., DUARTE, B. and LASSALLE, S., K-bounded polynomials, Math. Proc. R. It. Acad. 9 8 A (1998), 159-171.

ICE] CARANDO, D. and ZALDUENDO, I., A Hahn-Banach theorem for integral polynomials, Proc. Amer. Math. Soc. 127 (1999), 241-250.

[ D D ] DfAZ, J. C. and DINEEN, S., Polynomials on stable spaces, Ark. Mat. 36 (1998), 87-96.

[D] D I N E E N , S., Complex Analysis in Locally Convex Spaces, Math. Studies 57, North-Holland, Amsterdam, 1981.

[FGL] FERRERA, J., G6MEZ, J. and LLAVONA, J., On completion of spaces of weakly continuous functions, Bull. London Math. Soc. 15 (1983), 260-264.

[GI] GODEFROY, G. and IOCHUM, B., Arens-regularity on Banach algebras and geometry of Banach spaces, J. Funct. Anat. 80 (1988), 47-59.

[GJL] GUTD~RREZ, J., JARAMILLO, J. and LLAVONA, J., Polynomials and geometry of Banach spaces, Extracta Math. 10 (1995), 79-114.

[KR] KIRWAN, P. and RYAN, R., Extendibility of homogeneous polynomials on Ba- nach spaces, Proc. Amer. Math. Soc. 126 (1998), 1023-1029.

[M] MUJICA, J., Complex Analysis in Banach Spaces, Math. Studies 120, North- Holland, Amsterdam, 1986.

[ R y ] RYAN, R., Dunford-Pettis properties, Bull. Acad. Palon. Sci. Sdr. Sci. Math. 2T (I979), 373-379.

[S] STEGALL, C., Duals of certain spaces with the Dunford-Pettis property, Notices Amer. Math. Soc. 19 (1972), A799.

IT 1 TOMA, E., Aplicaqoes holomorfas e polinomios ~--continuos, Thesis, Universi- dade Federal do Rio de Janeiro, Rio de Janeiro, 1993.

[ ~ l ] ULGER, A., Weakly compact bilinear forms and Arens-regulaxity, Proc. Amer.

Math. Soc. 101 (1987), 697-704.

[Z1] ZALDUENDO, I., A canonical extension for analytic functions on Banach spaces, Trans. Amer. Math. Soc. 320 (1990), 747-763.

(12)

354 Silvia Lassaile and Ignacio Zalduendo:

To what extent does the dual Banach space E ' determine the polynomials over E?

[Z2]

ZALDUENDO, I., Extending p o l y n o m i a l s - - a survey, PubL Dep. Andlisis Mat., Univ. Complut. Madrid, Secc. 1, No. 41, 1998.

Received August 10, 1998 Silvia Lassalle

Departamento de Matemhtica (FCEN) Universidad de Buenos Aires

Pab. I - Ciudad Universitaria (1428) Buenos Aires

Argentina

Ignacio Zalduendo

Depaxtamento de Economla y MatemAtica Universidad de San Andr6s

CC 35 (1644) Victoria Buenos Aires

Argentina