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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 ${\parallel ·\parallel }_{+p}$ given on ${l}_{p}$ by ${\parallel 𝐱\parallel }_{+p}={\left[{{\Sigma }}_{n=0}^{\infty }\left({\left({x}_{n}\vee 0\right)}^{p}\right)\right]}^{1/p}$, where $𝐱:={\left({x}_{n}\right)}_{n\in \omega }$. We also obtain some results on completeness and compactness of these spaces.

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