Duality and quasi-normability for complexity spaces

Abstract

<p>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,) <sup>ω</sup>. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual.</p> <p>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*<sub>E</sub>, <sub>B*</sub> ) is biBanach, where B*<sub>E</sub> = {f : E Σ<sup>∞</sup><sub>n=0</sub> 2<sup>-n</sup>( V ) } and <sub>B*</sub> = Σ<sup>∞</sup><sub>n=0</sub> 2<sup>-n</sup> 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,)<sup>ω</sup> but also in the general case that it is a subspace of F<sup>ω</sup> 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.</p>