The generalized decomposition theorem in Banach spaces and its applications
type: Item
instance of: scholarly article
title: The generalized decomposition theorem in Banach spaces and its applications
zbMATH Open document ID: 1067.46009
author: Wen Song
author: Zhengjun Cao
published in: Journal of Approximation Theory
publication date: 2004-10-28T00:00:00Z
review text: A Hilbert space can be decomposed as the product of a closed subspace \(K\) with its orthogonal complement. An analogous result is true if \(K\) is a closed convex cone and the role of the complement of \(K\) is held by the polar cone of \(K\). In [Appl. Math. Lett. 11, No. 6, 115--121 (1998; Zbl 0947.46012) and Field Inst. Commun. 25, 77--93 (2000; Zbl 0971.46004)], \textit{Y. Alber} proved a form of a decomposition of a reflexive, strictly convex, smooth Banach space with respect to a closed convex cone. The authors present a simple proof of Alber's decomposition, and then prove a new decomposition theorem for such Banach spaces. The decompositions are related to the best approximation operator, and the authors characterize when the generalized projection is the best approximation operator.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 2111898
zbMATH Keywords: metric projection
zbMATH Keywords: duality mapping
zbMATH Keywords: convex cones
zbMATH Keywords: reflexivity
zbMATH Keywords: strict convexity
zbMATH Keywords: smoothness
zbMATH Keywords: basic variational principle
zbMATH Keywords: Moreau decomposition theorem
zbMATH Keywords: generalized decomposition theorem
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
full work available at URL: j.jat.2004.06.002
OpenAlex ID: W2049102775
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DOI: 10.1016/J.JAT.2004.06.002
description: scientific article; zbMATH DE number 2111898