The range of a projection of small norm in \(\ell^ n_ l\)
type: Item
instance of: scholarly article
title: The range of a projection of small norm in \(\ell^ n_ l\)
zbMATH Open document ID: 0545.46014
DOI: 10.1007/BF02761679
author: M. Zippin
published in: Israel Journal of Mathematics
publication date: 1981-01-01T00:00:00Z
review text: A Banach space X is called a \(P_{\lambda}\)-space if whenever X is contained in Y there is a projection P of Y onto X with \(\| P\|<\lambda\). It is not known, in general, what the \(P_{\lambda}\) spaces are for \(\lambda>1\). The conjecture that such k-dimensional spaces are close, using the Banach-Mazur distance, to \(\ell_{\infty}\) is equivalent by duality to showing that for any projection with norm \(<\lambda\) of \(\ell^ n_ 1\) onto a k-dimensional subspace, its range is close to \(\ell^ k_ l.\) The author addresses this question in the case \(\lambda\) is ''near'' 1 having proved the theorem: There exist a positive \(\epsilon_ 0\) and a positive function \(\phi\) (\(\epsilon)\) defined for \(0<\epsilon<\epsilon_ 0\) with \(\lim_{\epsilon \to 0}\phi(\epsilon)=0\) such that if k and n, \(n>k>0\), are integers and Q is a projection of \(\ell^ n_ l\) onto a k- dimensional subspace E with \(\| Q\| \leq 1+\epsilon\) then there is an integer \(h\geq k(1+\phi(\epsilon))\) and an h dimensional subspace F of E with \(d(F,\ell^ n_ l)\leq 1+\phi(\epsilon)\). Moreover, there is a projection P of \(\ell^ n_ l\) onto F with \(\| P\| \leq 1+\phi(\epsilon).\)
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46M10
zbMATH DE Number: 3867996
zbMATH Keywords: \(P_{\lambda}\)-space
zbMATH Keywords: Banach-Mazur distance
reviewed by: J. H. M. Whitfield
MaRDI profile type: Publication
cites work: A remark on finite-dimensional $P_{λ}$-spaces
cites work: Projections in Normed Linear Spaces
cites work: Banach Spaces with the Extension Property
cites work: A Theorem of the Hahn-Banach Type for Linear Transformations
description: scientific article; zbMATH DE number 3867996