The bidual of the space of polynomials on a Banach space

Jaramillo, Jesús Á.; Prieto, Ángeles; Zalduendo, Ignacio

doi:10.1017/s0305004197001898

Cited by 9 publications

(10 citation statements)

References 8 publications

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“…The closure of P f ( k E) will be denoted by P c ( Moreover, P wb ( k E) = P wsc ( k E) if and only if E does not contain l 1 (see [14]), while P c ( k E) = P wb ( k E) provided that E * has the approximation property. The Aron-Berner extension (see [4], [16]) provides us with a linear map…”

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confidence: 99%

Norm-attaining polynomials and differentiability

Ferrera

2002

Studia Math.

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confidence: 99%

Norm-attaining polynomials and differentiability

Ferrera

2002

Studia Math.

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“…the spaces E where, for every m, a natural map b m X P m E HH À3P m E HH is an isomorphism. In [33] the map b m was studied and conditions for Q-reflexivity were given. We refer to [20, 23, 24, 40 and 41] for further information about Q-reflexivity.…”

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confidence: 99%

Duality and reflexivity in spaces of polynomials

Jaramillo

Moraes

2000

Archiv der Mathematik

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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

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“…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).…”

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confidence: 99%

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

Carando¹,

Lassalle²

2004

Ark. Mat.

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Abstract. We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on E and F, we show that the spaces of X-vahed n-homogeneous polynomials and analytic functions of bounded type on E and F are isomorphic whenever X is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on E and F are isomorphic without conditions on the involved spaces.

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The bidual of the space of polynomials on a Banach space

Cited by 9 publications

References 8 publications

Norm-attaining polynomials and differentiability

Norm-attaining polynomials and differentiability

Duality and reflexivity in spaces of polynomials

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

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