Summability characterization for B-convex and super reflexive spaces
type: Item
instance of: scholarly article
title: Summability characterization for B-convex and super reflexive spaces
zbMATH Open document ID: 0558.46011
author: K. P. R. Sastry
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1984-01-01T00:00:00Z
review text: In this paper the following was proved: Theorem: A Banach space X is B- convex (resp. super reflexive) if and only if for every double sequence \(\{x^ m_ n\}\) in the unit ball of X, there is a lower triangular (resp. lower triangular and sign ordered) absolutely convex matrix \(T=(C_{mn})\) such that \[ (I)\quad \limsup\quad \| t_ m\| <1\quad where\quad t_ m=\sum^{m}_{n=1}C_{mn}x^ m_ n. \] It was observed that (I) can be replaced by \[ (II)\quad \lim \inf \quad \| t_ m\| <1. \] In the B-convex case, it was further observed that (II) can be replaced by \[ (III)\quad \lim \inf \quad \| t_ m\| =0. \] The paper concludes with an open problem: Can we replace (II) by (III) in the super reflexive case? Subsequently, this problem is answered in the affirmative independently by \textit{S. V. R. Naidu} [Dept. of Applied Mathematics, A.U.P.G. Extension Centre, Nuzvid, 521 201, India] in a private talk and \textit{J. R. Partington} [Fitzwilliam College, Cambridge, CB30DG, England] in a private communication.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3890142
zbMATH Keywords: B-convex
zbMATH Keywords: super reflexive
zbMATH Keywords: absolutely convex matrix
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3890142