A constructive approach to minimal projections in Banach spaces
type: Item
instance of: scholarly article
title: A constructive approach to minimal projections in Banach spaces
zbMATH Open document ID: 0596.41051
DOI: 10.1016/0021-9045(85)90107-8
author: David L. Motte
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: Let X be a Banach space, Y a finite dimensional subspace of X and \({\mathcal P}(X,Y)\) be the family of all projections of X onto Y. The projection \(P\in {\mathcal P}(X,Y)\) will be called a minimal projection if \(\| P\| \leq \| Q\|\) for all \(Q\in {\mathcal P}(X,Y)\). In order to compute the norm of the minimal projections, the author considers the sequences of finite dimensional ''approximating subspaces'' of X, \(\{X_ m\}\) and \(\{Y_ m\}\) and the corresponding minimal projections \(P_ m: X_ m\to Y_ m\) such that \(\lim_{m\to \infty}\| P_ m\| =\| P\|.\) One shows that such sequences can be constructed in many important Banach spaces including C[a,b], \(L^ p[a,b]\) and \(\ell^ p\) for \(1\leq p<\infty\) and \(c_ 0\). In such spaces, sequences of projections onto Y can also be constructed, having minimal projections as cluster points (in the strong operator topology). As an application, \(\inf \{\| Q\|: Q\in {\mathcal P}(X,Y)\}\) is computed in some particular cases.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 3960215
zbMATH Keywords: minimal projection
zbMATH Keywords: approximating subspaces
zbMATH Keywords: Banach spaces
zbMATH Keywords: cluster points
zbMATH Keywords: operator topology
zbMATH Keywords: application
reviewed by: Ioan Şerb
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90107-8
OpenAlex ID: W2081055985
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description: scientific article; zbMATH DE number 3960215