Duality and quasi-normability for complexity spaces

Abstract

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.

Weak condition for generalized f-weakly Picard mappings on partial metric spaces

“…In the context of computer science where a computable function can also be proved to be a contraction, the partial metric extension of the contraction fixed point theorem can be used to prove that the unique fixed point, which is the programs output, will be totally computed [18]. Further applications of partial metrics to problems in theoretical computer science were discussed in [11,12,25,26,28,29].…”

The complexity space of partial functions: a connection between complexity analysis and denotational semantics

“…In the last years the interest in dual complexity spaces has increased and they have been studied in depth ( [24], [9], [10], [11], [18], [17], [20], [16], [21]). …”

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

“…This study was motivated, in great part, by the fact that quasi-normed linear spaces provide suitable mathematical models in the theory of computational complexity (see [4,5,20]). …”