## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

#### 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

https://projecteuclid.org/euclid.bbms/1161350696

Digital Object Identifier
doi:10.36045/bbms/1161350696

Mathematical Reviews number (MathSciNet)
MR2307690

Zentralblatt MATH identifier
1126.46015

#### 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