Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

Romaguera, Salvador; Sanchis, Manuel

doi:10.1006/jath.1999.3439

Cited by 67 publications

(62 citation statements)

References 16 publications

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“…We start this section giving the definitions of a quasi-norm and of a quasinormed space in the sense of [5], [6] and [21] (see [4] for the related notion of a nonsymmetric norm).…”

Section: Bibanach Function Spacesmentioning

confidence: 99%

Duality and quasi-normability for complexity spaces

Romaguera

Schellekens

2002

Appl. Gen. Topol.

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Abstract. The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0, +∞) ω . Several quasi-metric properties of the complexity space were obtained via the analysis of its dual. We here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E, . ) is a biBanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (B * E , . B * ) is biBanach, where B *< +∞}, and f B * = ∞ n=0 2 −n f (n) . We deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and Smyth complete, not only in the case that this dual is a subspace of [0,+∞) ω but also in the general case that it is a subspace of F ω where F is any biBanach normweightable space. We also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. Finally, we investigate completeness of the quasi-metric of uniform convergence and of the Hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively.2000 AMS Classification: 54E50, 54E15, 54C35, 46E15.

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“…We start this section giving the definitions of a quasi-norm and of a quasinormed space in the sense of [5], [6] and [21] (see [4] for the related notion of a nonsymmetric norm).…”

Section: Bibanach Function Spacesmentioning

confidence: 99%

Duality and quasi-normability for complexity spaces

Romaguera

Schellekens

2002

Appl. Gen. Topol.

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“…In the last years the study of real-valued semi-Lipschitz functions dened on a T 0 quasi-pseudo-metric space has received a certain attention [11,12,16,18]. In particular, it was shown in [16] that the set of realvalued semi-Lipschitz functions dened on a T 0 quasi-pseudo-metric space (X, d) that vanish at a point x 0 ∈ X can be structured as a normed cone.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, it was shown in [16] that the set of realvalued semi-Lipschitz functions dened on a T 0 quasi-pseudo-metric space (X, d) that vanish at a point x 0 ∈ X can be structured as a normed cone. Applications of semi-Lipschitz functions to questions on best approximation, global attractors on dynamical systems, and concentration of measure can be found in [13,16], [17] and [22], respectively.…”

Section: Introductionmentioning

confidence: 99%

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

2008

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“…The arguments will derive, in general, from Measure Theory and they will provide the convergence of certain sequences of functions. As an example we mention that the subset of non decreasing functions in the Orlicz space L φ (0, 1) is proximinal in the sense that for each function f ∈ L φ (0, 1) there is a non decreasing function g ∈ L φ (0, 1) that minimizes the gap function Another interesting situation is when X is quasi-metric space (see for example [21], where a characterization of best approximation result is proved in this context).…”

Section: Introductionmentioning

confidence: 99%

A topological approach to Best Approximation Theory

et al. 2004

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Abstract. The main goal of this paper is to put some light in several arguments that have been used through the time in many contexts of Best Approximation Theory to produce proximinality results. In all these works, the main idea was to prove that the sets we are considering have certain properties which are very near to the compactness in the usual sense. In the paper we introduce a concept (the wrapping) that allow us to unify all these results in a whole theory, where certain ideas from Topology are essential. Moreover, we do not only cover many of the known classical results but also prove some new results. Hence we prove that exists a strong interaction between General Topology and Best Approximation Theory. AMS Classification:41A65, 54-99

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Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

Cited by 67 publications

References 16 publications

Duality and quasi-normability for complexity spaces

Duality and quasi-normability for complexity spaces

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

A topological approach to Best Approximation Theory

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