## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

Jesús Rodríguez-López and Salvador Romaguera

#### 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