Lower semicontinuity concepts, continuous selections, and set valued metric projections
type: Item
instance of: scholarly article
title: Lower semicontinuity concepts, continuous selections, and set valued metric projections
zbMATH Open document ID: 1001.41021
author: F. Deutsch
author: V. Indumathi
author: Aldric L. Brown
author: Petar S. Kenderov
published in: Journal of Approximation Theory
publication date: 2002-12-15T00:00:00Z
review text: Metric projections onto finite dimensional subspaces of a Banach space are always upper semicontinuous, but generally they are not lower semicontinuous. Lower semicontinuity guarantees other basic properties of the metric projection. For example, such metric projections always admit a continuous selection. This paper distills the fundamental continuity concepts of metric projections that have such important implications. It investigates a number of semicontinuity properties, the relationship between them and their implications. For example, a geometric characterization is given of those normed linear spaces such that the metric projection onto each of its one dimensional subspaces has a continuous selection. Then the spaces \(C_O(T)\) and \(L_1(\mu)\) that have this property are determined.
reviewed by: Daniel Wulbert
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1750343
zbMATH Keywords: best approximation
zbMATH Keywords: Banach spaces
zbMATH Keywords: continuous functions
zbMATH Keywords: \(L_p\)-spaces
MaRDI profile type: Publication
full work available at URL: jath.2001.3654
OpenAlex ID: W1971337797
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DOI: 10.1006/JATH.2001.3654
description: scientific article; zbMATH DE number 1750343