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Journal ArticleDOI

Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

TL;DR: In this paper, a graph defined by the Birkhoff-James orthogonality relation in normed spaces is studied, and it is shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually BJ orthogonal.
About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 2021-10-01. It has received 4 citations till now. The article focuses on the topics: Normed vector space & Vector space.
Citations
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Book ChapterDOI
01 Jan 2022

7 citations

Posted Content
TL;DR: In this paper, the authors construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers.
Abstract: Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and non-smooth direct sums, in arbitrary dimensions, we construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers among them.

4 citations

Journal ArticleDOI
TL;DR: In this paper , the notion of geometric structure spaces of Banach spaces equipped with closure space structure was introduced, and it was shown that the nonlinear equivalence of these spaces based on Birkhoff-James orthogonality induces homeomorphisms between geometric structures.

3 citations

Journal ArticleDOI
TL;DR: In this article , the authors compare Birkhoff-James orthogonality with some other orthogonalities, present its properties and its applications, and review the characterizations of the relation.
Abstract: We present Birkhoff–James orthogonality from historical perspectives to the current development. We compare it with some other orthogonalities, present its properties and its applications, and review the characterizations of Birkhoff–James orthogonality in classical Banach spaces like $$\mathbb B(\mathcal {H})$$ , C ∗-algebras, Hilbert C ∗-modules, or the space of rectangular matrices normed with Schatten norms. We also present the results on characterizations of preservers of Birkhoff–James orthogonality and, by devising a directed graph of the relation, show that in smooth spaces it can completely determine the norm up to (conjugate) linear isometry. Most, though not all, of the results that we state are supplied with (sketches of) the proof.
References
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Book
01 Nov 1970
TL;DR: In this paper, the authors propose an approximation of sous-espace lineaire de dimension finie, which is a lineaire lineaire of the dimension of the element d'ensemble.
Abstract: espace lineaire norme # espace metrique # meilleure approximation # sous-espace lineaire # sous-espace lineaire de dimension finie # sous-espace lineaire ferme de codimension finie # element d'ensemble # element d'ensemble non-lineaire

756 citations

Journal ArticleDOI
TL;DR: The notion of orthogonality was introduced in this paper, which is a generalization of the notion of homogeneous homogeneous elements to normed linear spaces, and has been studied extensively in the literature.
Abstract: The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are equivalent to this and can be generalized to normed linear spaces, preserving the property that every twodimensional linear subset contain nonzero orthogonal elements. The definition which will be used here (Definition 1.2) has the added advantage of being closely related to the theories of linear functionals and hyperplanes. The theory and applications of this orthogonality have been organized in the following sections, which are briefly outlined: 1. Fundamental definitions. An element x is orthogonal to an element y if and only if j|x+kyjj > ||x|l for all k. This orthogonality is homogeneous, but is neither symmetric nor additive. 2. Existence of orthogonal elements. An element x of a normed linear space is orthogonal to at least one hyperplane through the origin, while for elements x and y there is at least one number a for which ax+y Ix or ||ax+yj| is minimum (Theorems 2.2-2.3). 3. Orthogonality in general normed linear spaces. The limits N?(x; y) =+lim,+.?|lnx+yil jjnxjj =liml0?o [||x+hyjj -lixil ]/hexistandsatisfyweakened linearity conditions. Also, x Iax+y if and only if N_(x; y) < -a||x||

495 citations

Journal ArticleDOI
TL;DR: The second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) is presented in this article, where the authors discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowowski spaces, and bisectors as well as Voronoi diagrams in Minkowsky spaces.

305 citations