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Sequence spaces and asymmetric norms in the theory of computational complexity. (English) Zbl 1063.68057
Summary: In 1995, M. Schellekens [Proc. MFPS 11, Electronic Notes in Theoretical Computer Science 1, 211–232 (1995; Zbl 0910.68135)] introduced the complexity (quasi-metric) space as a part of the development of a topological foundation for the complexity analysis of algorithms. Recently, S. Romaguera and M. Schellekens [Topology Appl. 98, 311–322 (1999; Zbl 0941.54028)] have obtained several quasi-metric properties of the complexity space which are interesting from a computational point of view, via the analysis of the so-called dual complexity space.
Here, we extend the notion of the dual complexity space to the \(p\)-dual case, with \(p > 1\), in order to include some other kinds of exponential time algorithms in this study. We show that the dual \(p\)-complexity space is isometrically isomorphic to the positive cone of \(l_p\) endowed with the asymmetric norm \(\|\cdot\|_{+p}\) given on \(l_p\) by \(\| \mathbf x\|_{+p} = [{\Sigma}_{n=0}^{\infty}((x_n \vee 0)^p)]^{1/p}\), where \(\mathbf x := (x_n)_{n\in {\omega}}\). We also obtain some results on completeness and compactness of these spaces.

MSC:
68Q25 Analysis of algorithms and problem complexity
46A45 Sequence spaces (including Köthe sequence spaces)
54E15 Uniform structures and generalizations
54C35 Function spaces in general topology
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