Some results in representable Banach spaces
type: Item
instance of: scholarly article
title: Some results in representable Banach spaces
zbMATH Open document ID: 0663.46010
DOI: 10.5565/PUBLMAT_32188_06
author: Miguel A. Canela
published in: Publicacions Matemàtiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41023
review text: A linear normed space X is called universally representable [\textit{G. Godefroy} and \textit{M. Talagrand}: Isr. J. Math. 41, 321-330 (1982; Zbl 0498.46016)] if it is analytic (i.e. it is a continuous image of the irrationals) whenever it is endowed with pointwise convergence on the points of an arbitrary countable norming set of its dual. If the property holds for a certain countable norming set, the space X is called representable. Firstly, the author extends a result of Godefroy and Talagrand concerning the characterization of the universally representable Banach spaces: If X is a representable Banach space, then the following properties are equivalent: (i) X is universally representable; (ii) X does not contain any copy of \(\ell_ 1([0,1]);\) (iii) \(X^*\) is \(w^*\)-angelic; (iv) \(B_{X^*}\) is \(w^*\)-sequentially compact; (v) \(B_{X^*}\) is \(w^*\)-Rosenthal compact; (vi) X can be renormed such that the closed unit ball of \(X^*\) is \(w^*\)-separable and \(w^*\)-Rosenthal compact; (vii) for any measurable space (\(\Omega\),\(\Sigma)\), the totally scalarly measurable functions f: \(\Omega\) \(\to X\) are also scalarly measurable. The main result of the paper is concerned to the stability in forming tensor products: Let X, Y be two Banach spaces. Then: (i) If X, Y both have countable type, X\({\hat \otimes}_{\epsilon}Y\) has countable type. (ii) If X, Y are representable, X\({\hat \otimes}_{\epsilon}Y\) is representable, and it is universally representable if and only if X, Y are both universally representable. Consequently, applications to the space C(K,X) are given.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 4084316
zbMATH Keywords: injective tensor product
zbMATH Keywords: universally representable
zbMATH Keywords: representable Banach space
zbMATH Keywords: \(w^*\)-angelic
zbMATH Keywords: \(w^*\)-sequentially compact
zbMATH Keywords: \(w^*\)-Rosenthal compact
zbMATH Keywords: totally scalarly measurable functions
zbMATH Keywords: stability in forming tensor products
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4084316