Duality and quasi-normability for complexity spaces

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 : ω → E | ∑∞ n=0 2 −n(‖f(n)‖∨‖−f(n)‖) < +∞}, 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 Fω 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. AMS (2000) Subject classification: 54E50, 54E15, 54C35, 46E15.