\(E^{\prime }\) and its relation with vector-valued functions on \(E\)
type: Item
instance of: scholarly article
title: \(E^{\prime }\) and its relation with vector-valued functions on \(E\)
zbMATH Open document ID: 1058.46024
DOI: 10.1007/BF02385480
author: Daniel Carando
author: Silvia Lassalle
published in: Arkiv för Matematik
publication date: 2004-09-28T00:00:00Z
review text: This paper investigates the following question: If the duals of the Banach spaces \(E\) and \(F\) are isomorphic (isometric) and \(X\) is a Banach space, does this mean that the spaces of \(X\)-valued \(n\)-homogeneous polynomials on \(E\) and \(F\) are isomorphic (isometric)? A Banach space \(E\) is said to be symmetrically regular if all symmetric linear operators from \(E\) into \(E'\) are weakly compact. When \(E\) and \(F\) are symmetrically regular, \(s: E'\to F'\) is an isomorphism (resp. isometry) and \(X\) is a Banach space, the authors observe that the mapping \({\bar s}(P)=\overline{P}\circ s'\circ J_F\) defines an isomorphism (resp. isometry) from the space of \(X\)-valued \(n\)-homogeneous polynomials on \(E\) onto the space of \(X''\)-valued \(n\)-homogeneous polynomials on \(F\). Moreover, when \(X=W'\) is a dual Banach space the range of \(\overline{s}\) is contained in the space of \(X\)-valued \(n\)-homogeneous polynomials on \(F\). When \(s\) is an isomorphism (isometry) of \(E'\) onto \(F'\) and \(X\) is a Banach space, the authors show that \(\overline{s}\) is an isomorphism between the subspaces of \(X\)-valued finite-type (resp. nuclear, approximable, regular, integral, extendible, \(n\)-homogeneous polynomials which are weakly continuous on bounded sets) on \(E\) and spaces of \(X\)-valued finite-type (resp. nuclear, approximable, regular, integral, extendible, \(n\)-homogeneous polynomials which are weakly continuous on bounded sets) on \(F\). When \(E\) does not contain a copy of \(\ell_1\), the same is true for the spaces of vector-valued weakly sequentially continuous polynomials. The authors show that when \(E\) and \(F\) are symmetrically Arens regular and \(W'\) is a dual Banach space, then \({\mathcal H}_b(E,W')\) and \({\mathcal H}_b(F,W')\) are isomorphic when \(E'\) and \(F'\) are isomorphic and that \({\mathcal H}_b(B_E,W')\) (resp. \({\mathcal H}^\infty(B_E,W')\)) and \({\mathcal H}_b(B_E,W')\) (resp. \({\mathcal H}^\infty (B_E,W…
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 2103849
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: morphisms
zbMATH Keywords: subspaces of polynomials
MaRDI profile type: Publication
cites work: A Hahn-Banach extension theorem for analytic mappings
cites work: \(Q\)-reflexive Banach spaces
cites work: Weakly continuous mappings on Banach spaces
cites work: Uniform factorization for compact sets of operators
cites work: Polynomials on dual-isomorphic spaces
cites work: Extendible polynomials on Banach spaces
cites work: Duality in spaces of nuclear and integral polynomials
cites work: A Hahn-Banach theorem for integral polynomials
cites work: Polynomials on stable spaces
cites work: Strongly \(p\)-summing multilinear operators.
cites work: Integral holomorphic functions
cites work: Q4257163
cites work: Extension of multilinear mappings on Banach spaces
cites work: Q4487406
cites work: Injective factorization of holomorphic mappings
cites work: Duality and reflexivity in spaces of polynomials
cites work: The bidual of the space of polynomials on a Banach space
cites work: To what extent does the dual Banach space \(E'\) determine the polynomials over \(E\)?
cites work: Integral mappings between Banach spaces
cites work: Q3431289
full work available at URL: https://doi.org/10.1007/bf02385480
OpenAlex ID: W2093192471
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description: scientific article; zbMATH DE number 2103849