Orthogonality and smoothness induced by the norm derivatives
type: Item
instance of: scholarly article
title: Orthogonality and smoothness induced by the norm derivatives
zbMATH Open document ID: 1471.46015
author: Debmalya Sain
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2021-06-30T00:00:00Z
review text: This paper is concerned with variations on the orthogonality (similar to the classical Birkhoff-James orthogonality) in terms of certain derivative notions involving the norm on a linear space \(X\). In general, these notions of orthogonality are not symmetric, transitive, nor the orthogonal complement a linear space. It is generally known that some form of smoothness is related to the linearity of the orthogonal complement. One of the orthogonalities considered for non-zero vectors \(x,y \in X\) is: \(x\perp_{\rho}y\) if and only if for a unit vector \(f \in X^\ast\) that attains its norm at \(\frac{x}{\|x\|}\), there is a \(g\) with similar properties such that \(f(y)=-g(y)\). As in the classical case, it turns out that \(x\) is \(\rho\)-smooth if and only if the orthogonal complement is a subspace of codimension \(1\).
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 47L05
zbMATH DE Number: 7365218
zbMATH Keywords: norm derivatives
zbMATH Keywords: orthogonality in normed spaces
zbMATH Keywords: smoothness
zbMATH Keywords: linear operators
MaRDI profile type: Publication
full work available at URL: s13398-021-01060-0
OpenAlex ID: W3169327740
cites work: Q3654874
cites work: Characterizations of inner product spaces
cites work: Orthogonality in linear metric spaces
cites work: On a \(\rho\)-orthogonality
cites work: Q4821622
cites work: \(M\)-ideals in Banach spaces and Banach algebras
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Q3797696
cites work: Birkhoff-James orthogonality and smoothness of bounded linear operators
cites work: Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces
cites work: On the norm attainment set of a bounded linear operator and semi-inner-products in normed spaces
cites work: Operator norm attainment and inner product spaces
cites work: A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space
cites work: BIRKHOFF ORTHOGONALITY IN CLASSICAL -IDEALS
cites work: Linear mappings preserving \(\rho \)-orthogonality
cites work: Semi-smooth points in space of operators on Hilbert space
cites work: Characterization of linear similarities through functional equation and mappings preserving orthogonalities
DOI: 10.1007/S13398-021-01060-0
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description: scientific article; zbMATH DE number 7365218