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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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

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

First available in Project Euclid: 20 October 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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