The Dual Complexity Space as the Dual of a Normed Cone

In [15] M. Schellekens introduced the complexity (quasi-metric) space as a part of the research in Theoretical Computer Science and Topology, with applications to the complexity analysis of algorithms. Later on, S. Romaguera and M. Schellekens ([13]) introduced the so-called dual complexity (quasi-metric) space and established several quasi-metric properties of the complexity space via the analysis of th e dual. These authors also proved in [14] that actually the dual complexity space C can be modeled as a norm-weightable cone whose induced quasi-metric is Smyth complete. This fact suggests the existence of deep connections between a general theory of (dual) complexity spaces and Asymmetric Functional Analysis. These connections have been recently explored in [3], [4] and [8]. In particular, it was proved in [3] that the so-called dual p-complexity space C∗ p , with 1 ≤ p < ∞, is isometrically isomorphic to the positive cone of the classical Banach space lp. The space C 1 is exactly the dual complexity space, and thus it is isometrically isomorphic to the positive cone of the Banach space l1 of all absolutely summable real sequences. Here, we continue the analysis of the structure of the dual complexity space C. We show that it is the dual space of the positive c one of the Banach space c0 of all real sequences converging to zero, and that its dual space is the positive cone of the Banach space l∞ of all bounded real sequences. Furthermore, the dual space of C p , 1 < p < ∞, is C∗ q where 1/p+ 1/q = 1. These results extend to this setting well-known theorems of the classical theory of Functional Analysis.