Bulletin of the Belgian Mathematical Society - Simon Stevin

Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric

Jesús Rodríguez-López and Salvador Romaguera

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Abstract

If $A$ is a (nonempty) bounded convex subset of an asymmetric normed linear space $(X,q),$ we define the closedness of $A$ as the set \textrm{cl}$% _{q}A\cap \mathrm{cl}_{q^{-1}}A,$ and denote by $CB_{0}(X)$ the collection of the closednesses of all (nonempty) bounded convex subsets of $(X,q).$ We show that $CB_{0}(X),$ endowed with the Hausdorff quasi-metric of $q,$ can be structured as a quasi-metric cone. Then, and extending a classical embedding theorem of L. H\"{o}rmander, we prove that there is an isometric isomorphism from this quasi-metric cone into the product of two asymmetric normed linear spaces of bounded continuous real functions equipped with the asymmetric norm of uniform convergence.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 3 (2006), 551-562.

Dates
First available in Project Euclid: 20 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1161350696

Digital Object Identifier
doi:10.36045/bbms/1161350696

Mathematical Reviews number (MathSciNet)
MR2307690

Zentralblatt MATH identifier
1126.46015

Subjects
Primary: 54B20: Hyperspaces 54C25: Embedding 54C35: Function spaces [See also 46Exx, 58D15]

Keywords
The Hausdorff quasi-metric asymmetric normed linear space quasi-metric cone bounded convex subset closedness isometric isomorphism

Citation

Rodríguez-López, Jesús; Romaguera, Salvador. Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 3, 551--562. doi:10.36045/bbms/1161350696. https://projecteuclid.org/euclid.bbms/1161350696


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