Journal ArticleDOI
Duality mapping and birkhoff orthogonality
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In this article, the authors established some properties of Birkhoff orthogonality in terms of duality mapping, using a slight extension of a Rockafellar result, and obtained a new proof of a result earlier established by Blanco and Turnsek concerning the linear operators preserving Birkoff orthOGonality.Abstract:
In this note we establish some properties of Birkhoff orthogonality in terms of duality mapping. Particularly, using a slight extension of a Rockafellar result we obtain a new proof of a result earlier established by Blanco and Turnsek concerning the linear operators preserving Birkhoff orthogonality. Mathematics Subject Classication 2010: 46B20, 46B10, 46C50.read more
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Orthogonality in Generalized Minkowski Spaces
TL;DR: In this article, the authors combine functional analytic and geometric viewpoints on approximate Birkhoff and isosceles orthogonality in generalized Minkowski spaces which are finite-dimensional vector spaces equipped with a gauge.
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Operators preserving orthogonality are isometries
TL;DR: In this article, it was shown that every operator from a real Banach space into itself preserving orthogonality is an isometry multiplied by a constant, and that the same is true for any operator from an operator from another operator into itself.
Book ChapterDOI
On Exact and Approximate Orthogonalities Based on Norm Derivatives
Ali Zamani,Mahdi Dehghani +1 more
TL;DR: In this paper, a survey of orthogonality relations in normed linear spaces related to norm derivatives is presented, focusing on fundamental properties of norm derivatives, differences and connections between these orthogonsality types, and geometric results and problems closely related to them.
References
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Book ChapterDOI
Normed Linear Spaces
TL;DR: A(D) as discussed by the authors is a function space with norm ∥ ∥ [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: ∥ x - y ∥ ≦ e}.
Book
Convexity and optimization in Banach spaces
Viorel Barbu,Theodor Precupanu +1 more
TL;DR: In this article, the authors propose a method to solve the problem of convex control problems in Banach spaces. But this method is not suitable for functional analysis.Convex Functions and Convex Programming
Journal ArticleDOI
Normed Linear Spaces. By M. M. Day. Pp. 139. DM 28. 1958. (Springer, Berlin)
Journal ArticleDOI
On the maximal monotonicity of subdifferential mappings.
TL;DR: In this article, it was shown that the subdifferential of a lower semicontinuous proper convex function on a Banach space is a maximal monotone operator, as well as a maximal cyclically monotonous operator.