The Dual Complexity Space as the Dual of a Normed Cone

Abstract

In [M. Schellekens, The Smyth completion: A common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, vol. 1 (1995), 23 pages] 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 ([S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999), 311–322]) 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 [S. Romaguera, M. Schellekens, Duality and quasi-normability for complexity spaces, Appl. Gen. Topology 3 (2002), 91–112] 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 [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11], [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, The supremum asymmetric norm on sequence spaces: a general framework to measure complexity distances, in: Proceedings of the Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 12 pages] and [M. O'Keefe, S. Romaguera, M. Schellekens, Norm-weightable Riesz spaces and the dual complexity space, in: Proceedings of the Second Irish Conference on the Math ematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 17 pages]. In particular, it was proved in [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11] that the so-called dual p-complexity space Cp∗, with 1⩽p<∞, is isometrically isomorphic to the positive cone of the classical Banach space lp. The space C1∗ 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 Cp∗, 1<p<∞, is Cq∗ where 1/p+1/q=1. These results extend to this setting well-known theorems of the classical theory of Functional Analysis.