Duality and quasi-normability for complexity spaces

By Salvador Romaguera, Michel P. Schellekens. 2002

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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* E = {f : E Σ ∞ n=0 2 -n ( V ) } and B* = Σ ∞ n=0 2 -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.

Subfields: Geometry and Topology, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/4198067cc052b8f8d8595d707133647341a686cc

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In Geometry and Topology

In Management Science and Operations Research

Citations in the corpus

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