Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces
type: Item
instance of: scholarly article
title: Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces
zbMATH Open document ID: 0947.46012
DOI: 10.1016/S0893-9659(98)00112-8
published in: Applied Mathematics Letters
publication date: 2000-10-31T00:00:00Z
review text: Let \(B\) be a real uniformly convex and uniformly smooth (reflexive) Banach space, \(B^*\) its conjugate space, \(J:B\to B^*\) the duality mapping determined by the equalities \(\langle Jx,x\rangle = \|Jx\|\|x\|=\|x\|^2,\) and let \(W(x,\xi) = \|x\|^2 - 2\langle Jx,x \rangle + \|\xi\|^2 \). The functional \(W\), called the Lyapunov functional, is positive and it is closely related to the Young-Fenchel transform. In [Lect. Notes Pure Appl. Math. 178, 15-50 (1996; Zbl 0883.47083)], the author defined and studied the properties of generalized projection operators \(\Pi_\Omega\) and \(\pi _\Omega=J^*\Pi_\Omega\) where, for a closed convex subset \(\Omega\) of \(B\), \(\Pi_\Omega\) is defined by \(\Pi_\Omega = \widehat x,\quad \widehat x\) being the solution of the minimization problem \(W(x,\widehat x) =\inf _{\xi \in \Omega} W(x,\xi).\) If \(B\) is a Hilbert space then \(Jx = x\), \(W(x,\xi) = \|x-\xi\|^2\) and \(\Pi_\Omega\) becomes the usual metric projection operator on \(\Omega.\) As it was shown by the author, \textit{A. N. Iusem} and \textit{M. V. Solodov} [J. Convex Anal. 4, No. 2, 235-255 (1997; Zbl 0895.90150)], the generalized projection operators have nice properties which make them very effective, in contrast to the metric projection operators, in the study of approximation and optimization problems in Banach spaces. The aim of the present paper is to study the properties of the generalized projection operators in the particular case when \(\Omega\) is a closed convex cone \(K\) of the Banach space \(B\). One proves decomposition theorems for arbitrary elements, both in primary and dual spaces, and one establishes Pythagorean-type rules, \(\|x\|^2 = \|\widehat x\|^2 + W(x,\widehat x),\) results which were known previously only in Hilbert spaces.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 46N10
zbMATH DE Number: 1369487
zbMATH Keywords: duality mapping
zbMATH Keywords: generalized projection operators
zbMATH Keywords: dual and polar cones
zbMATH Keywords: variational principles
zbMATH Keywords: Lyapunov functionals
zbMATH Keywords: Young-Fenchel transform
zbMATH Keywords: Pythagorean-type rules
Wikidata QID: Q114120812
author: Yakov Alber
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
cites work: Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications
cites work: Q4881794
cites work: Q4382546
cites work: Geometry of Banach spaces. Selected topics
cites work: On the moduli of convexity and smoothness
cites work: Q3845569
cites work: Q5550647
cites work: Q5516366
full work available at URL: s0893-9659(98)00112-8
OpenAlex ID: W2081107090
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description: scientific article; zbMATH DE number 1369487