On balancedness and D-completeness of the space of semi-Lipschitz functions

Romaguera, Salvador; Sánchez-Álvarez, José Manuel; Sanchis, Manuel

doi:10.1007/s10474-007-7159-2

Cited by 5 publications

(7 citation statements)

References 18 publications

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“…We also prove that it is complete in the sense of D. Doitchinov. These results generalize those obtained in [18] because, in our study, the asymmetric normed space does not necessarily satisfy the T 1 axiom. Moreover, we provide a class of asymmetric normed spaces whose dual cones are right K-sequentially complete.…”

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confidence: 87%

Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space

Alegre

2022

Results Math

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We obtain some quasi-metric properties of the dual cone of an asymmetric normed space. Thus, we prove that it is balanced, and hence its topology is completely regular. We also prove that it is complete in the sense of D. Doitchinov. These results generalize those obtained in [18] because, in our study, the asymmetric normed space does not necessarily satisfy the T 1 axiom. Moreover, we provide a class of asymmetric normed spaces whose dual cones are right K-sequentially complete. Finally, we represent an arbitrary asymmetric normed space as a function space by using the unit ball of its dual space.

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confidence: 87%

Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space

Alegre

2022

Results Math

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“…In [15] some properties of the "normed cone" (d-SLip 0 Y, •| d ) are presented. Similar properties in the case of d−semi-Lipschitz functions on a quasi-metric space with values in a quasi-normed space (space with asymmetric norm) are discussed in [16], [17]. For more information concerning other properties of quasi-metric spaces, see also [7], [13].…”

Section: The Cone Of Semi-lipschitz Functionsmentioning

confidence: 76%

Best uniform approximation of semi-Lipschitz functions by extensions

Mustăţa

2007

J. Numer. Anal. Approx. Theory

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In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function \(F\) defined on an asymmetric metric space \((X,d),\) by the elements of the set \(\mathcal{E}_{d}(\left. F\right\vert _{Y})\) of all extensions of \(\left.F\right\vert _{Y}\) \((Y\subset X),\) preserving the smallest semi-Lipschitz constant. It is proved that this problem has always at least a solution, if \((X,d)\) is \((d,\overline{d})\)-sequentially compact, or of finite diameter.

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“…Among these we mention Dolzhenko and Sevastyanov with the papers quoted above, Sevastyanov [160], a quasi-metric on the space SLip ρ,q,0 (X, Y ). The following completeness result was proved in [143].…”

Section: 2mentioning

confidence: 91%

“…The properties of the spaces of semi-Lipschitz functions were studied by Romaguera and Sanchis [149,151] and Romaguera, Sánchez-Álvarez and Sanchis [143]. The paper by Mustȃţa [117] is concerned with the behavior of the extreme points of the unit ball in spaces of semi-Lipschitz functions An important result in the study of Lipschitz functions on metric spaces is the extension of Lipschitz functions, usually known as Kirszbraun's extension theorem, see, for instance, the book [172].…”

Section: 3mentioning

confidence: 99%

Functional Analysis in Asymmetric Normed Spaces

Cobzaş

2013

Frontiers in Mathematics

199

171

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The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space X is not a linear space, but merely a convex cone in the space of all linear functionals on X. Due to this fact, a careful treatment of the duality problems (e.g. reflexivity) and of other results as, for instance, the extension of fundamental principles of functional analysis -the open mapping theorem and the closed graph theorem -to this setting, is needed. Contents 1. Introduction: 1.1 Quasi-metric spaces and asymmetric normed spaces; 1.2 The topology of a quasisemimetric space; 1.3 Quasi-uniform spaces.2. Completeness and compactness in quasi-metric and in quasi-uniform spaces: 2.1 Various notions of completeness for quasi-metric spaces; 2.2 Compactness, total bondedness and precompactness; 2.3 Completeness in quasi-uniform spaces; 2.4 Baire category.3. Continuous linear operators between asymmetric normed spaces: 3.1 The asymmetric norm of a continuous linear operator; 3.2 The normed cone of continuous linear operators -completeness; 3.3 The bicompletion of an asymmetric normed space; 3.4 Open mapping and closed graph theorems for asymmetric normed spaces; 3.5 Normed cones; 3.6 The w ♭ topology of the dual space X ♭ p ; 3.7 Compact subsets of an asymmetric normed space; 3.8 The conjugate operator, precompact operators between asymmetric normed spaces and a Schauder type theorem; 3.9 Asymmetric moduli of smoothness and rotundity; 3.10 Asymmetric topologies on normed lattices. 4. Linear functionals on an asymmetric normed space: 4.1 Some properties of continuous linear functionals; 4.2 Hahn-Banach type theorems; 4.3 The bidual space, reflexivity and Goldstine theorem. 5. The Minkowski functional and the separation of convex sets: 5.1 The Minkowski gauge functional -definition and properties; 5.2 The separation of convex sets; 5.3 Extremal points and Krein-Milman theorem.6. Applications to best approximation: 6.1 Characterizations of nearest points in convex sets and duality; 6.2 The distance to a hyperplane; 6.3 Best approximation by elements of sets with convex complement; 6.4 Optimal points; 6.5 Sign-sensitive approximation in spaces of continuous or integrable functions.7. Spaces of semi-Lipschitz functions: 7.1 Semi-Lipschitz functions -definition, the extension property, applications to best approximation in quasi-metric spaces; 7.2 Properties of the cone of semi-Lipschitz functions -linearity, completeness. ReferencesThis research was supported by Grant CNCSIS 2261, ID 543.Remark 1.1. Since the terms "quasi-norm", "quasi-normed space" and "quasi-Banach space" are already "registered trademarks" (see, for instance, th...

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On balancedness and D-completeness of the space of semi-Lipschitz functions

Cited by 5 publications

References 18 publications

Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space

Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space

Best uniform approximation of semi-Lipschitz functions by extensions

Functional Analysis in Asymmetric Normed Spaces

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