Duality and reflexivity in spaces of polynomials

Abstract

We first study the reflexivity of the space P m EY F of continuous m-homogeneous polynomials between Banach spaces E and F. Then, in a more general way, we obtain conditions under which the spaces P m EY F HH and P m E HH Y F HH are canonically isomorphic.Introduction. In the last few years there has been an increasing interest in the study of reflexivity and related properties for spaces of polynomials on Banach spaces. This has been motivated in part by the connection with the corresponding properties of spaces of holomorphic mappings and also with the problem of extending entire mappings to the second dual.After the pioneering work of Ryan [39] about reflexivity of the spaces of polynomials, the first example of an infinite-dimensional Banach space E for which the space P m E of all continuous m-homogeneous polynomials on E is reflexive for every m was obtained by Alencar, Aron and Dineen in [5], choosing E to be the Tsirelsons space. Further examples of spaces with this property were given in [11]. These spaces are called polynomially reflexive by Farmer [21], who extended these previous examples by using the theory of spreading models. Some of the results in [21] have been sharpened in [25, 27 and 28]. In all these works, as well as in [3], the connection between reflexivity of P m E and properties such as weak sequential continuity or approximability of polynomials is stressed. Further refinements along this line were given in [37], where reflexivity results are obtained under a very mild approximation property. Most of these papers present applications to spaces of holomorphic functions and holomorphic germs. This is the case of [23] where, making more precise the work of Prieto [38], Galindo, Maestre and Rueda characterized the bidual of certain spaces of holomorphic functions.For vector-valued mappings, Holub [31] showed that if both E and F are reflexive and one of them has the Approximation Property, then the space LEY F of continuous linear mappings from E into F is reflexive if, and only if, every element of LEY F is compact. Later on, the case of vector-valued polynomials was considered in [6, 4, 7, 8 and 26]. In [4], Alencar characterizes the reflexivity of P m EY F when the reflexive spaces E and F have both the Approximation Property. In Section 1 we are going to show that the same

E′ and its relation with vector-valued functions on E

“…We describe the vector-valued version of the inclusion of (P(kE))" into P(kE") studied in [2] and [20], which was introduced in [19]. First, define, for zCE", the mapping e~: P(kE; X)~X" by e~(P)=P(z).…”

Polynomials, Symmetric Multilinear Forms and Weak Compactness

“…A related ques-tion, the reflexivity of the space of continuous n-homogeneous polynomials, has been studied by several authors (see [4,12,10,13]). …”

Hilbert spaces of analytic functions of infinitely many variables

“…It is known [10] that the space P( n X) of all continuous polynomials (with sup-norm) on a Banach space X with the approximation property is reflexive if and only if all polynomials from P( n X) are weakly sequentially continuous, and this is equivalent to the fact that every P ∈ P( n X) is norm attaining. The next theorem shows that for subspaces of P( n X) the situation is different.…”