Quasi-metric properties of complexity spaces

Romaguera, Salvador; Schellekens, Michel

doi:10.1016/s0166-8641(98)00102-3

Cited by 107 publications

(74 citation statements)

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“…It is shown in [22] that the dual complexity space (C * F , d . C * ) is not precompact and, thus, not totally bounded, in general.…”

Section: By Theorem 24 (Bmentioning

confidence: 99%

“…F ), consider the complexity space (C F∞ , d C ) and the dual complexity space (C * F , d C * ). Then, as in [22], we may define an isometry Ψ : C F∞ → C * F by Ψ(f ) = 1/f for all f ∈ C F∞ . Combining this fact with the propositions and theorems proved in this section, we obtain, among other, the following results:…”

Section: By Theorem 24 (Bmentioning

confidence: 99%

“…by the isometry Ψ : C * → C, defined by Ψ(f ) = 1/f (see [22]). Thus, via the analysis of its dual, several quasi-metric properties of (C, d C ), in particular Smtyh completeness and total boundedness, are studied in [22].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, via the analysis of its dual, several quasi-metric properties of (C, d C ), in particular Smtyh completeness and total boundedness, are studied in [22]. A motivation for the use of the dual instead of the original complexity space is the fact that the dual is mathematically somewhat more appealing, since d C * is "derived" from the restriction to R + of the quasi-metric u defined above.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, it is possible to carry out the complexity analysis of algorithms based on the dual complexity space. In fact, the dual complexity space has the advantage that it respects the interpretation usually given to the minimum ⊥ in semantic domains (see [22,Section 4]). …”

Section: Introductionmentioning

confidence: 99%

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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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“…It is shown in [22] that the dual complexity space (C * F , d . C * ) is not precompact and, thus, not totally bounded, in general.…”

Section: By Theorem 24 (Bmentioning

confidence: 99%

Section: By Theorem 24 (Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Duality and quasi-normability for complexity spaces

Romaguera

Schellekens

2002

Appl. Gen. Topol.

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Functional Analysis in Asymmetric Normed Spaces

Cobzaş

2013

Frontiers in Mathematics

210

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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An Application of Generalized Complexity Spaces to Denotational Semantics via the Domain of Words

Llull-Chavarría

Valero

2009

Language and Automata Theory and Applications

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Quasi-metric properties of complexity spaces

Cited by 107 publications

References 8 publications

Duality and quasi-normability for complexity spaces

Duality and quasi-normability for complexity spaces

Functional Analysis in Asymmetric Normed Spaces

An Application of Generalized Complexity Spaces to Denotational Semantics via the Domain of Words

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