Functionals generated dually by minimal projectors, and criteria for Banach spaces to be Hilbert
type: Item
instance of: scholarly article
title: Functionals generated dually by minimal projectors, and criteria for Banach spaces to be Hilbert
zbMATH Open document ID: 0619.46015
DOI: 10.1007/BF01157542
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let B be a real Banach space and \(0\neq x\in B\). Assume there is a unique hyperplane \(D_ x\) such that for each hyperplane D in B with \(x\not\in D\| P(D,x)\| >\| P(D_ x,x)\|\), where P(D,x) is the projection onto D along sp(x). For any \(z\in B\) set \(\phi_ x(z)=a\| x\|\), where \(z=ax+P(D_ x,x)z.\) Theorem 1. Let B be a reflexive Banach space such that for each hyperplane D in \(B^*\) there exists a unique minimal projection onto D. Then the functional \(\phi_ x\) is well-defined for each nonzero \(x\in B.\) Theorem 2. Let B be a uniformly smooth and strictly convex Banach space (Note that such space satisfies the hypothesis of theorem 1.) Then B is isometrically isomorphic to a Hilbert space iff \(\| x\| \phi_ x(y)=\| y\| \phi_ y(x)\) for all nonzero x,y\(\in B\).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 4004906
zbMATH Keywords: minimal projections
zbMATH Keywords: characterization of Hilbert spaces
zbMATH Keywords: uniformly smooth and strictly convex Banach space
author: W. Odyniec
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01157542
OpenAlex ID: W2069400907
description: scientific article; zbMATH DE number 4004906