Extending polynomials on Banach spaces—a survey

Ignacio Zalduendo

A la memoria de Pucho Larotonda, maestro y amigo

Introduction

Throughout, , , and will be complex Banach spaces, a subspace of . The question which gave rise to the material reviewed here was first posed by Dineen [D1] in relation to holomorphic completeness and in the context of more general locally convex spaces. It is the following: when can a continuous homogeneous polynomial be extended to ? Several answers—none complete—have been given for varying hypotheses on the spaces and the polynomials involved, and several applications have appeared. We attempt here to unify the existing approaches to the problem, and to point out the common ingredients in the solutions.

It is clearly not possible in general to extend polynomials to larger spaces, in fact this is true even of linear functions: the identity cannot be extended to , for is non-complemented in . When is the complex field and is linear there is of course the Hahn-Banach theorem, but we shall see that in the non-linear case, there are scalar-valued polynomials which cannot be extended.

The positive results in this matter can be loosely grouped into two main categories: those providing linear extension morphisms by which all polynomials can be extended to some larger space, and Hahn-Banach type extensions applicable to some polynomials and extending those to all larger spaces. We concentrate on the first type of result in sections 1 and 2, and on the second type in section 3.

We restrict our attention to polynomials over Banach spaces, although many results have been obtained regarding the extension of holomorphic functions on Banach or other locally convex spaces ([Bol], [CM], [GGM2], [Kh1], [Kh2], [Mo2], [Mo3], [MV]). Also, many applications have appeared which we do not go into here: to holomorphic completeness [D1], to geodesics [DT], polarization constants [LR], reflexivity of spaces of polynomials [JPZ] [AD], the spectra of algebras of analytic functions over Banach spaces [ACG1] [AGGM], integral representation of analytic functions [PZ], and orthogonally additive polynomials ([BLLl], [PgV], [CLZ]).

In the first section we consider the Aron-Berner extension. We begin with the Arens product in a commutative Banach algebra, a very specific extension of a bilinear function, but nevertheless an extension in which some of the main points of more general extensions already appear, such as lack of symmetry and the notion of regularity. We then define and study the Aron-Berner extension, an extension of polynomials from a Banach space to its bidual. In the second section, we consider extensions from to . Here all solutions stem from the existence of a continuous linear extension morphism for linear forms , a condition obviously stronger than Hahn-Banach, and not satisfied in all cases. Section 3 is devoted to Hahn-Banach type extensions. We are naturally drawn to 'linearization' of polynomials, and thus to preduals of spaces of polynomials. The space of 'extendible' polynomials is considered also.

This survey has been brewing, on and off, for about a decade. It started out as a talk prepared for the Conference on Polynomials and Holomorphic Functions on Locally Convex Spaces, in University College Dublin in September of 1994, and then lay dormant for a couple of years until the Departamento de Análisis Matemático of the Universidad Complutense de Madrid invited the author to lecture on these topics in the winter of 1997, thus providing the oportunity for extensive rewriting. A first version appeared as a preprint [Z3] in the series Publicaciones del Departamento de Análisis Matemático of the Complutense. More recently, the author has lectured on these matters in a Seminar at the Universidad de Buenos Aires, and so had the oportunity to add more recent material —particularly to section 3— and make these notes up to date. It is a pleasure to thank my colleagues at UCD, UCM, and UBA for their interest and support. Also, many participants at the Conference in Dublin, and many of those present at the Madrid and Buenos Aires lectures have helped to mold these notes with their comments, their questions, and in more than a few cases, their answers. In this respect, it is a pleasure to acknowledge the contribution of J. M. Ansemil, R. Aron, F. Bombal, C. Boyd, D. Carando, V. Dimant, S. Dineen, J. A. Jaramillo, J. G. Llavona, A. Peris, A. Prieto, R. Ryan, S. Lassalle, and A. Tonge.

A word of caution. In writing these notes I have had in mind my students more than my colleagues. Thus, the result has been more an introduction than a survey. The experts will find some of their theorems missing, their proofs changed, their points of view distorted. The choice of material has been personal, and probably guided by aesthetic sense more than anything else. Also, in many places, the absence of a reference is not due to my originality, but to my ignorance.

Before going on to the first section of this paper, we succintly present the basic definitions and notation of the theory of polynomials on Banach spaces. For a truly comprehensive text, see [D4].

A -homogeneous polynomial is a function which can be written as where is a -linear function from to . When is symmetric, it can be recovered from the corresponding polynomial by any of several polarization formulas. Thus, there is a one-to-one correspondence between -homogeneous polynomials and symmetric -linear functions.

The 'uniform' norm of can be defined as

where is the closed unit ball of . Continuity of (and of the corresponding symmetric -linear function ) is equivalent to finiteness of . The space of -valued continuous -homogeneous polynomials, with the uniform norm, is a Banach space which we will denote by , or simply by when is the scalar field.

We will need to consider several more restricted classes of scalar-valued polynomials, each of which gives rise to a vector subspace of .

Surely the simplest kind of polynomials over a Banach space are the polynomials of finite type, those which may be written as , where each is a continuous linear form over . Slightly larger is the class of nuclear polynomials, made up of those which may be represented (non-uniquely) as , with . These form a non-closed vector subspace of (although they do form a Banach space in the 'nuclear' norm, i.e. the infimum of over all possible representations of ). The closure of the space of nuclear (or also of the finite-type polynomials) in the uniform norm is the space of approximable polynomials. Larger still are the spaces of weakly continuous polynomials (those which are weakly continuous over all bounded subsets of ) and weakly sequentially continuous polynomials. We thus have the chain of vector subspaces of

Reverse inclusions sometimes hold, highlighting the interplay between the study of polynomials and Banach space theory: if has the approximation property [AP], if has the Dunford-Pettis property [Ry1], while holds if and only if contains no copy of [FGLl]. Another important class of polynomials is that of integral polynomials: those which may be represented by a regular Borel measure over the unit ball of (considered with the weak topology) in the following way

Like the nuclear polynomials, these form a Banach space with their own particular 'integral' norm (the infimum of the total variations of the representing measures), but are a non-closed subspace of in the uniform norm. We have the inclusions

with when has the Radon-Nikodym property [A] (for example, if is reflexive).

The only trivially extendible scalar-valued polynomials are the finite-type and the nuclear polynomials. These can be extended to any larger space by simply using the Hahn-Banach theorem to extend their component linear forms . We shall see that the integral polynomials are also extendible but that there are, in general, approximable polynomials which cannot be extended.

§1. The Aron-Berner extension

In 1951 R. Arens ([A1], [A2]) found a way of extending the product of a Banach algebra to its bidual in such a way that this bidual became itself a Banach algebra. We restrict our presentation here to commutative Banach algebras , for in the sequel we will be interested in polynomials, and thus in symmetric multilinear forms.

Suppose then that , , and . The elements and are defined by setting

Now, given and in the bidual of , define the product of and by

for each . It is easily seen that this product extends the product in . Also, we observe the following two facts which will have analogues in the more general Aron-Berner extension.

i) The map is weak-continuous of its first variable: Indeed, if is weak-converging to , then . ii) All elements of commute with all elements of : Take and . Then for any

but one checks that is just .

In general, the map in i) is not continuous of the second variable, and the Arens product is not commutative. In fact we will see in the next section that these two properties are equivalent. To see that the product is not commutative, we present the following example [Z2] (see also [Re]).

Example 1.1.

Let be the commutative Banach algebra with the convolution product

and define the subspace of Now use Hahn-Banach to extend and to elements and of . These do not commute. To see this, take and observe that Thus and . Analogously . Now For more on the Arens product, see [IL], [DH], and [GI].

The process applied to the map can be carried out on an arbitrary continuous bilinear function

as follows: fix and ; then we obtain an element of . This mapping is bilinear and continuous, producing . If we repeat the process again we obtain , and when repeated once more, Now if instead of considering Banach algebras one takes , a Banach space, , and a symmetric bilinear form on , the construction produces an extension of to .

The same extension can be defined in other ways. Here are two:

a) Say and are elements of the bidual, and choose nets and converging to and in the weak topology of . Now put

This definition does not depend on the particular choice of nets, and produces the same extension as above. Note that is not, in general, weak-continuous in the second variable, but only in the first.

b) Define the associated symmetric linear map by . Its bitranspose corresponds to a continuous bilinear extension of

Similar constructions can be carried out for -linear forms. Note that in all cases there is a choice involved in the ordering (of the limits, of and ). It turns out that even though is symmetric, the extended form need not be symmetric. In fact one has the following equivalences, with and as above.

Proposition 1.2. ([DH], [Rp]) The following are equivalent:

1. is symmetric.
2. is separately weak-continuous in both variables.
3. is weakly compact.

Proof. i) implies ii): is always weak-continuous in the first variable, for transpose maps are weak to weak-continuous. Thus symmetry implies weak-continuity in the other variable as well.

ii) implies iii): Fix any . Weak-continuity of in the variable says that is actually in the canonical inclusion of in . Thus and is weakly compact by Gantmacher's theorem [HP].

iii) implies i): We denote the canonical inclusion of in by and that of in by . First, note that since is symmetric, . Indeed, for any and in ,

This in turn implies that for any , : for all , we have

Thus, if is weakly compact we obtain : Since , and separates points of , we need only see that for and , . But this is . Thus we have,

and is symmetric. □

It is worth noting that elements of always commute with those of , even when is not symmetric. Thus for any choice of and .

If the equivalent conditions in the proposition hold for all symmetric , is said to be symmetrically regular, and if all (symmetric or otherwise) operators are weakly compact, is regular. As Example 1.1 shows, is not symmetrically regular (and hence not regular). The operator corresponding to Example 1.1 is given by

where the sign of is taken as . It is easily seen that is not weakly compact: any subsequence of (with in th place) which converges weakly does so to , but for all . So the image of the unit ball is not weakly sequentially compact.

Another example of such an operator [ACG1] is given by

and yet another [Re] by

On the other hand, spaces are regular [GI], as are all -algebras.

We continue with some results on symmetric regularity, as found in [AGGM].

Proposition 1.3. is regular if and only if all operators in , , , are weakly compact.

Proof. Just note that one may write as

thus . □

An immediate consequence is that is regular if and only if is regular. Also, if is non-reflexive, is non-regular.

Proposition 1.4.

1. If is symmetrically regular, then is regular.
2. is regular if and only if it is symmetrically regular.

Proof. i) Take , any continuous operator, and define by

where . Then

Thus is symmetric, and therefore it, and , are compact.

ii) This is now easy from part i) and the fact that is regular if and only if is. □

We now define the Aron-Berner extension of a continuous -homogeneous scalar-valued polynomial defined over the Banach space . We choose to give a degree- version of the first definition mentioned above, but note that the second is also easily adapted to this case. We first extend -linear forms:

Given and , define

by . Here denotes the linear form obtained from by fixing variables (note that by symmetry it does not matter where these variables are fixed). The map is linear, continuous and of norm . Now, given we may define the extended -linear form by Here we have identified with the scalar field. The extended mapping need not be symmetric. We have in fact extensions, one for each choice of ordering in . Note however that for we do have , and that is weak-continuous in the first variable [Z1]. These properties are analogous to those of the Arens product.

When is symmetrically regular, variables in can be permuted in pairs even in the extensions of -linear forms, so that if is symmetrically regular the extension of any symmetric -linear form is itself symmetric.

We define the Aron-Berner extension of to the bidual of by setting

where is the unique symmetric continuous -linear form such that for every .

It must be stressed that this extension is quite algebraic in nature. Indeed, the formal manipulations involved can be carried out with no regard to topology whatsoever. It is certainly not an extension by continuity, and need not be weak-continuous (consider the polynomial over ).

However the Aron-Berner extension respects most of the usual classes of continuous polynomials, as we shall see below. Also, a relation with the weak-topology in is present, though perhaps more subtly than one would expect, as the next two theorems show.

First, there is the following characterization of the Aron-Berner extension [Z1] in terms of first-order differentials.

Theorem 1.5. [Z1] If and , then if and only if

1. For each , is weak-continuous, and
2. For each and converging weak to , . □

Condition b) is somewhat odd, but examples show that need not be weak-continuous for and that condition a) alone does not guarantee that . Indeed, let and such that (with notation as in Example 1.1). Then the Aron-Berner extension of is . However, since is not weak-continuous in the second variable, choose converging weak to with ; let and . Then weak, but . Also, if , then is weak-continuous for all , but .

One consequence of the above characterization is the linearity of the Aron-Berner extension mapping . Indeed, , and for all the differential has the same continuity properties as each summand. The same holds for scalar multiples of .

Thus, the Aron-Berner extension is a one-to-one linear map

The continuity of this map is a simple matter, for the norm of the extended polynomial verifies: . However, more can be said. Davie and Gamelin have proved [DG] the following theorem which shows that in fact the Aron-Berner map is an isometry onto its image.

Theorem 1.6. . In fact, given and a polynomial , there is a bounded net weak-converging to and for which .

Proof. Let be the symmetric -linear form associated with . Fix in the unit ball of , and . For any natural number (to be fixed later) choose, using Goldstine's theorem, an element in the unit ball of such that

Then choose in the unit ball of such that and go on until one has in the unit ball of such that whenever , then We then have, for , or (considering that is symmetric) for any different indexes Now let . where in the first sum we group all terms with no repeated indexes, and in the second all remaining terms. Since each term with non-repeated indexes is smaller than , we have but the term on the right can be made small by taking large; i.e. the larger the number of indexes that one uses, the smaller the chance of having repeated indexes in a group of of them. Thus can be as close as as required. □

A polynomial may in general admit many other norm-preserving extensions to , but there are a few uniqueness results. For example, 2-homogeneous norm-attaining polynomials on have unique norm-preserving extension to [ABC].

In view of the Davie-Gamelin theorem, one can consider the Aron-Berner extension as an inclusion map

A few comments on the preservation of the usual classes of polynomials through the Aron-Berner extension: Clearly finite type, nuclear, and approximable polynomials are preserved. Less clear but nevertheless true is the fact that integral polynomials over are mapped into integral polynomials over . We have:

Theorem 1.7. [CZ1] The Aron-Berner extension of an integral polynomial over is an integral polynomial over . Furthermore, if contains no copy of , and is a measure representing , then

Note that this integral makes no sense if contains a copy of , for need not be a -measurable function over [H]. Also, an analogue of the Davie-Gamelin theorem holds: the integral norm of is the same as that of . The same holds for the class of extendible polynomials (which we will present in §3). De Moraes has proved:

Theorem 1.8. [Mo1] The Aron-Berner extension of a weakly continuous polynomial over is a weakly continuous polynomial over . □

Another proof of this fact is possible through the characterization of weakly continuous polynomials as those which are continuous with respect to the seminorm

for some compact subset of [ALRR], [T]. Indeed, it has been shown [AG], [CDDL] that for any particular , such polynomials are preserved by the Aron-Berner extension.

The Aron-Berner extension does not, in general, preserve the class of weakly sequentially continuous polynomials, as the following example shows.

Example 1.9. Given any Banach space , consider the 2-homogeneous polynomial defined by . It is easily seen that weak sequential continuity of this polynomial is equivalent to having the Dunford-Pettis property.

Now consider a Banach space having the Dunford-Pettis property whose dual does not have it (say , [S]), and consider the weakly sequentially continuous polynomial

The Aron-Berner extension of this polynomial is

where is the restriction mapping (i. e., ). Thus if we restrict to the subspace , we obtain the polynomial If were weakly sequentially continuous, would be also, and hence would have the Dunford-Pettis property, contradicting our assumption.

A positive result in this matter is the following theorem of González and Gutiérrez (who actually prove a more general result).

Theorem 1.10. [GG] If has the Dunford-Pettis property, then the Aron-Berner extension preserves weakly sequentially continuous polynomials. □

Through the Aron-Berner extension, polynomials defined on can be evaluated at points of , and indeed at points of any even dual (, ,...) of . It is natural to ask then if evaluations at such points constitute new evaluations or if, for example, given there is an element which polynomials cannot distinguish from , that is for all such . Clearly, for there is such an element of ; it is , where is the canonical inclusion of in its bidual . The matter for is far more interesting, and has been studied by Aron, Galindo, García and Maestre, who found the following.

Theorem 1.11. [AGGM] Each evaluation at a point of is in fact an evaluation at a point of if and only if is symmetrically regular. □

For -valued polynomials , the extension process described above produces an -valued polynomial . It can be seen that if is weakly compact, then the image of is in , although (unlike the linear case) the converse is not true. However, even in situations in which is -valued, analogues of Theorem 1.5 and Theorem 1.6 hold [C1].

§2. Extension morphisms from to

Several particular cases have been studied ([AB], [DT], [GGMM], [LR], [Z1]) of situations in which all polynomials defined over a Banach space are extended to a certain larger Banach space . In all these instances there turns out to be an extension morphism. We present in this section a unified version of the above mentioned results, and several formulations equivalent to the existence of an extension morphism for polynomials.

We shall need the following result of Aron and Schottenloher [AS].

Theorem 2.1. Let and be arbitrary Banach spaces. Then is isomorphic to a complemented subspace of .

Proof. Fix and a norm-one linear functional such that . Define the 'inclusion' by (the product is point-wise). Also, define the 'projection' by . Note that , where is the symmetric -linear operator associated to , and that if ,

Thus . Indeed, for any , we have □

The existence of a linear extension mapping is equivalent to the existence of a linear extension mapping . We have the following theorem. The 'if' part is a reformulation of results in [GGMM] and [Ni], the 'only if' part is due to A. Peris.

Theorem 2.2. Let be a subspace of , and an arbitrary Banach space. Then there is an extension morphism if and only if there is an extension morphism

Proof. Suppose exists, and define and as in Theorem 2.1. Now let . is continuous, linear, and for and , :

(the last equality as in Theorem 2.1). Note also that in differentiating, we have used the fact that is in for any .

Now suppose that is given. We want to define , and proceed as in the definition of the Aron-Berner extension. Given and , define

by . Here denotes the linear form obtained from by fixing variables. Now, given we may define the extended -linear function by Thus if is a polynomial in , and its associated symmetric -linear function, set It is now easily checked that is a linear extension mapping from to . □

Note that the function cannot be defined (even non-linearly) if is not linear. Indeed, a non-linear would induce a which would fail to be -linear in the above definition. Also, note that the constructions and are not in general inverse of each other.

The fact that is a subspace of and an extension map makes an extension map; however, the construction can be carried through for any continuous linear map , giving rise to a continuous linear map

Note however, that need not be . The following example is taken from [LZ].

Example 2.3. :

If is a Banach space, consider (as in Example 1.9) the 2-homogeneous polynomial defined over by . Recall that the Aron-Berner extension of to is

where is the restriction (i.e., the transpose of the natural inclusion ). We will use the notation and over these spaces consider the polynomials and morphisms and given by ,. Also, we denote . We now check that .

First, calculate :

Now calculate : so using this we have Then thus if and only if , but this only happens for reflexive .

We now provide some examples of situations in which a linear extension map exists for linear operators, and thus also for polynomials. We begin with the case of -valued polynomials in the first three examples, and continue with scalar-valued polynomials in the rest.

Example 2.4. ( complemented in ):

This is clearly the simplest case for extension. If is a projector, any polynomial may be extended by setting . This extension is a morphism and comes from by the construction above.

Example 2.5. ([Z1]):

If , may be considered contained in through the inclusion mapping defined by . In this case one may define by .

Example 2.6.

If there is a linear extension map for linear forms , and is complemented in its bidual (this happens, for example if is a dual space), one may define as follows. First define by , and then , where is the projection . If exists but is not complemented in , one may define an extension morphism .

Example 2.7. (the Aron-Berner extension [AB]):

For , the Aron-Berner extension comes from the extension morphism , the canonical inclusion of in its bidual.

Example 2.8. ( in an 'local' ultrapower of [DT], [LR]):

Suppose and is finitely represented in in the following sense: for each finite-dimensional subspace of and each there is a isomorphism such that is the identity when restricted to and . In such a situation one can define a linear extension morphism in the following way: Consider the indexing set ordered by . The sets form a filter basis over . Let be any ultrafilter containing this basis. We have, for any ,

such that . For each , define Although the 's are not linear, is, and we may define is then a linear extension map, for if , . This induces an extension morphism , which is the extension in [LR] and [DT].

We now give more conditions equivalent to the existence of a linear extension morphism (and therefore equivalent to the existence of a linear extension morphism for polynomials). Other conditions may be found in [CGJ1] and in [D4, Prop. 6.18]. The implication ii) iii) is taken from [D3]. First, some definitions:

We say is finitely complemented in , if there is a constant such that for all finite-dimensional subspaces of , there is a projector such that , and for .

If is a subspace of we shall say, following Dineen [D3], that has the polynomial extension property if for all , each polynomial extends to a polynomial . Note that it is not required that these extensions define a linear morphism. denotes the ultrapower of corresponding to the ultrafilter .

Theorem 2.9. Let be a subspace of . Then the following are equivalent.

1. There is a linear extension morphism .
2. has the polynomial extension property.
3. is complemented in .
4. is finitely complemented in .
5. There is an ultrafilter and a continuous linear function such that for .

Proof. i) implies ii): Consider the dual of and the dual of in the usual way. Then

is a linear extension morphism from to . Thus by the preceding theorem (with ) every -homogeneous polynomial over extends to a -homogeneous polynomial over .

ii) implies iii): Define by . There is, by ii), a polynomial which extends . This polynomial extends by Aron-Berner to a polynomial over . We consider its restriction to . Now if , let

Then is an element of , and is continuous and linear. Now if and , the weak-closure of contains . Using the Davie-Gamelin theorem, construct a net weak-converging to and such that . Then Hence , and is a projection of onto .

iii) implies iv): If is a projector, take . For any finite-dimensional subspace of , is a finite-dimensional subspace of . By the local reflexivity principle [De] there is a linear operator

such that for , and if , . Then has norm , and for we have .

iv) implies v): Consider the indexing set

ordered by inclusion, let and be an ultrafilter on containing the 's. For each we have a projector of norm not exceeding . Define for each and let be defined as (the class of the family in the ultraproduct ). is linear -though the 's are not- and . Now if , for all , , so .

v) implies i): First define as . Now set . This is a linear extension morphism. Indeed, for all and , we have

□

Some comments are in order. First, in iii) of the theorem we consider the bitranspose of the inclusion to be the natural inclusion map of in . This may, in some cases, be misleading. Consider for example the canonical inclusion . Then it is easily checked that the inclusion is not the canonical one : indeed, let be non-zero, but such that Ker, and take such that . Then

The fact that and produce different extension morphisms is related to the evaluation problem considered in [AGGM].

Note ([CGJ1], [Ka]) also that from the point of view of homological algebra, while complementation of in is the splitting of the sequence

the properties in the Theorem are equivalent to the splitting of the dual sequence

Note also that one could write a 'norm sensitive' version of the theorem, putting ' norm preserving', , for any , and .

We close this section with an application of the procedure of Theorem 2.2. In [DD] Díaz and Dineen prove that if Banach spaces , have isomorphic duals, and has the Schur property and the approximation property, then and are isomorphic, i.e.: with the above hypotheses, the dual space determines the space of polynomials . If is an isomorphism, it seems natural to look into the map . As mentioned after Theorem 2.2, in general . However, in the presence of Arens regularity, the procedure is sufficiently well-behaved to produce the following result (proved independently in [CCG] and [LZ]).

Theorem 2.10. If and are symmetrically Arens-regular, and and are isomorphic (resp. isometric), then for any , and are isomorphic (resp. isometric). □

§3. Hahn-Banach type extensions

We now address the problem of when a continuous -homogeneous scalar-valued polynomial can be extended to any larger Banach space, regardless of whether other polynomials over the same space can be so extended and whether such extensions supply linear morphisms or not.

Following Kirwan and Ryan [KR], we shall say is extendible if it can be extended to a continuous polynomial over any Banach space containing as a subspace, and denote by the space of all such polynomials. We start with some immediate comments on extendible polynomials.

Remark 3.1.

All extendible polynomials are weakly sequentially continuous. Indeed, consider a subspace of (the space of continuous functions over the closed unit ball of with the weak topology), through the mapping where . If is an extendible polynomial over , it extends to a polynomial over . Since has the Dunford-Pettis property, every polynomial defined on it is weakly sequentially continuous [Ry1] and this forces to be weakly sequentially continuous as well.

Remark 3.2.

If is extendible, and is the symmetric linear operator associated to , then is weakly compact: Call the inclusion of in defined above, and let be an extension of . This means that , or equivalently, that there is a symmetric linear operator such that the following diagram commutes

Since is symmetrically regular, is weakly compact, and since is weak-to-weak continuous, is weakly precompact. Thus is weakly compact.

Remark 3.3.

From [KR]: If is a Hilbert space, and , then is extendible if and only if is nuclear: Suppose first that is extendible and, much as in the previous example, consider contained in . On extending one obtains the diagram

Since is an -space, and are 2-summing, and so is nuclear. Then is nuclear, and so is . The other implication is trivial, for all nuclear polynomials on any space are extendible.

We may now present a few concrete examples of non-extendible scalar-valued polynomials.

1) , given by cannot be extendible, for it is not weakly sequentially continuous: the unit basis in tends weakly to , but for any (also, is not nuclear).

2) , given by is not extendible. This is the polynomial corresponding to the non-weakly compact operator given in Example 1.1, so it cannot be extendible.

3) , given by is not extendible, for it is not nuclear (and is a Hilbert space). ; thus there are approximable polynomials which are not extendible.

There are, in general, many non-extendible scalar valued polynomials on a Banach space. There are some classes of polynomials, however, which are always extendible. All finite-type and all nuclear polynomials can be extended by simply applying the Hahn-Banach theorem to their component linear forms. It is also true that all integral polynomials over a Banach space are extendible [CZ1].

A problem related to the non-extendibility of polynomials and considered by P. Mazet in [Ma2] is the following: find the smallest constant such that for any hyperplane in every 2-homogeneous polynomial on can be extended to with the norm of the extension not exceeding . Clearly must be larger than one, for otherwise using a transfinite induction argument every would be extendible. Mazet finds the following bounds.

Theorem 3.4. For real spaces, . For complex spaces, . □

One way of extending polynomials defined on to would be to identify each with a linear form, and then apply the Hahn-Banach theorem to this linear form. Thus, we are interested in representing spaces of polynomials as dual spaces: say is algebraically isomorphic to a subspace of , and that is algebraically isomorphic to a subspace of , and that there is an linear inyection . Then if is topologically a subspace of , each (considered as a linear form over ) may be extended by Hahn-Banach to . In fact, any given polynomial will be extendible to if and only if, as a linear functional on it is continuous for the topology induced on by .

There have been several constructions of preduals of spaces of polynomials, and indeed, of spaces of holomorphic functions. In fact, the constructions usually produce more than just preduals. They produce solutions to the following problem: given a space of scalar-valued functions on the set , can one construct a space and a function of the same type as those of factoring all functions of and identifying each with a continuous linear form ? That is,

One can then identify algebraically and topologically with the dual of , and indeed, the space of -valued functions with the continuous linear operators . The existence of such a 'linearization' is strictly stronger than the mere existence of a predual.

Constructions of this type have been obtained for spaces of continuous homogeneous polynomials by R. Ryan [Ry2] through the use of symmetric tensor products. In the holomorphic setting, linearizations have been constructed for holomorphic functions by P. Mazet [Ma1] and J. Mujica and L. Nachbin [MN], for bounded holomorphic functions by J. Mujica [M2], and for holomorphic functions of bounded type by P. Galindo, D. García and M. Maestre [GGM1] and Mujica [M3] (see also [Boy]). A more abstract approach produces linearizations for a wide class of function spaces [CZ2].

We restrict our attention here to preduals of spaces of polynomials. Given , let us denote by the vector space of symmetric -fold tensor products of elements of . Its algebraic dual is the space of all (not necesarily continuous) -homogeneous polynomials over . Furthermore, if , there is a linear injection . By requiring more or less stringent continuity conditions of the elements of one may obtain all classes of polynomials over , in fact the following result holds [CZ1].

Proposition 3.5. If is any subspace of containing the finite type polynomials, is (algebraically) isomorphic to , where is a Hausdorff locally convex topology on . □

Note that C. Boyd and R. Ryan have proved that not all such subspaces of are dual Banach spaces [BoyR]. We illustrate the proposition above by presenting a few of the classical polynomial spaces as duals of with the topologies indicated. We denote a typical element of as a finite sum , where the 's are elements of (this representation is not unique).

a) is isometrically the dual of when this space is endowed with the projective tensor norm [Ry2], .

b) is isometrically the dual of , when endowed with the following norm (equivalent to the injective tensor norm):

c) is algebraically the dual of when endowed with the following locally convex topology [CZ1]. Consider, for each compact subset of , the following seminorm on : , and define the seminorm on by , where the infimum ranges over all possible representations of . If is the topology induced by all such seminorms, we obtain an algebraic isomorphism .

For more on tensor products of Banach spaces see [DF] and [Ry3].

We now review some results of Carando, Dimant, Kirwan, Pérez-García, Ryan, and Sevilla regarding the space of all extendible polynomials over . As a set, we have seen that is contained in the space of weakly sequentially continuous polynomials, and that it contains the space of integral polynomials, but not in general the approximable polynomials. Kirwan and Ryan have proved the following.

Theorem 3.6. Let be an extendible polynomial. There is then a constant such that for any Banach space containing , some extension of to has norm not larger than . □

In their proof they construct a Banach space which is an 'amalgamation' of all those which contain , and extend to this large space. It is therefore possible to define a norm on by setting

This norm is larger than the usual norm in , and is equivalent to it if and only if every polynomial is extendible. Also, is complete, and a predual of it is obtained by putting in the norm

where the 's range over all Banach spaces containing as a subspace.

In [C1], Carando takes a different approach to obtain a predual of . Starting with the inclusion defined in Remark 3.1, he considers

putting in the projective norm, and then considering in the norm induced by this map.

Apart from the map , another inclusion which has proved useful in the study of extendible polynomials is

has the metric extension property [DF], that is, given a linear map and , admits a norm-preserving extension to . The following results are adapted from [C1] and [CGJ1].

Theorem 3.7. Let be a scalar-valued -homogeneous polynomial over . Then the following are equivalent:

a) is extendible.

b) extends to .

b') For some compact set , there is a linear operator , and such that .

c) extends to .

c') For some set , there is a linear operator , and such that . □

In [CGJ1] the authors study extendibility of bilinear forms, and obtain, among other results:

Theorem 3.8. Let . Then the following are equivalent.

a) is extedible.

b) There exists an -space and operators and such that .

c) There exists a Hilbert space and two 2-summing operators such that . □

Results similar to those of Theorem 3.6 for vector-valued polynomials can be found in [C1]. There the following results for a -homogeneous polynomial are obtained.

Theorem 3.9. is extendible if and only if extends to . If is complemented in its bidual, is extendible if and only if extends to . □

An immediate corollary of b') above is

Corollary 3.10. If is extendible, and linear, then is extendible and . □

Thus, any restriction of an extendible polynomial is extendible. Also, if is extendible, so is its Aron-Berner extension. By c), and the fact that has the metric extension property, the norm of any extension of to any space can be bounded by the norm of any extension of to . Thus the 'extendible' norm defined above can also be expressed as

In [P] Pisier gave a counterexample to a well-known conjecture of Grothendieck [G] regarding the non-existence of non-nuclear locally convex spaces and such that the injective and projective tensor norms coincide on the tensor product . Pisier constructed a Banach space such that . Thus, over Pisier's space all 2-homogeneous polynomials are integral. The situation is very different for polynomials of degree higher than 2. Indeed, [Pg] has shown that any infinite-dimensional Banach space admits extendible non-integral polynomials of any degree higher that 3, and in [CD] Carando and Dimant close the gap by proving the same for any degree higher than 2. Their construction makes use of finite-dimensional estimates of Boas [Boa].

Any space of cotype 2 (this includes spaces for ) can be embedded in Pisier's space [P]. Thus if has cotype 2, all extendible 2-homogenesous polynomials over are integral. Indeed, an extension to Pisier's space will be integral, and thus also the original polynomial ([CGJ1], [C2]).

Regarding extendible polynomials on spaces, Carando, Dimant and Sevilla [C2] [CDS] have constructed on all spaces with extendible polynomials which are not integral.

Polynomials of degree two and bilinear forms have attracted particular attention in recent years. In [CGJ2] a homological approach is applied to the problem of extendibility of bilinear forms from to . Considerations of pushout and pullback diagrams produce the results:

Proposition 3.11. If for all the sequences

split, then all bilinear forms on extend to . If for all the sequences split, then all bilinear forms on extend to . □

Since some of these properties are known when is an -space, the authors concentrate on these and produce the following:

Theorem 3.12. If is an -space and , the following are equivalent.

a) All bilinear forms on extend to .

b) All exact sequences split.

c) All exact sequences split.

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Ignacio Zalduendo
Departamento de Matemática,
Universidad Torcuato Di Tella,
Miñones 2177 (C1428ATG),
Buenos Aires, Argentina
nacho@utdt.edu

Recibido: 12 de setiembre de 2005
Aceptado: 7 de agosto de 2006