Some results in representable Banach spaces (Q1114117)

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.