A fully convex Banach space which does not have the Banach-Saks property
type: Item
instance of: scholarly article
title: A fully convex Banach space which does not have the Banach-Saks property
zbMATH Open document ID: 0603.46019
DOI: 10.1016/0022-247X(86)90261-1
author: Pei-Kee Lin
author: Bor-Luh Lin
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: Let \(k\geq 2\) be an integer. A Banach space X is said to be fully k-convex if for every sequence \(\{x_ n\}\) in X, \(\lim_{n_ 1,...,n_ k\to \infty}(1/k)\| x_{n_ 1}+...+x_{n_ k}\| =1\) implies that \(\{x_ n\}\) is a convergent in X. Every uniformly convex space is fully 2-convex. A Banach space X is said to have the Banach-Saks property if for every bounded sequence \(\{x_ n\}\) in X, there is a subsequence \(\{y_ n\}\) of \(\{x_ n\}\) such that the Cesara mean, \(\{(1/n)(y_ 1+...+y_ n)\}\) of \(\{x_ n\}\) is convergent in X. It is known that every superreflexive space has the Banach-Saks property but there exists a reflexive space without the Banach -Saks property. In this paper, we show that there exists a fully 2-convex space which does not possess the Banach-Saks property. So the Milman's question whether every reflexive space is isomorphic to a fully 2-convex space is still open.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 3973581
zbMATH Keywords: fully k-convex
zbMATH Keywords: Banach-Saks property
zbMATH Keywords: Cesara mean
zbMATH Keywords: superreflexive space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3973581