December 2020 Index of symmetry and topological classification of asymmetric normed spaces
Mohammed Bachir, Gonzalo Flores
Rocky Mountain J. Math. 50(6): 1951-1964 (December 2020). DOI: 10.1216/rmj.2020.50.1951


Let X,Y be asymmetric normed spaces and Lc(X,Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X,Y) is not a vector space. The aim of this note is to give, using the Baire category theorem, a complete characterization on X and a finite dimensional Y so that Lc(X,Y) is a vector space. For this, we introduce an index of symmetry of the space X denoted c(X)[0,1] and we give the link between the index c(X) and the fact that Lc(X,Y) is in turn an asymmetric normed space for every asymmetric normed space Y. Our study leads to a topological classification of asymmetric normed spaces.


Download Citation

Mohammed Bachir. Gonzalo Flores. "Index of symmetry and topological classification of asymmetric normed spaces." Rocky Mountain J. Math. 50 (6) 1951 - 1964, December 2020.


Received: 28 February 2020; Revised: 10 June 2020; Accepted: 15 June 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.1951

Primary: 46A22‎ , 46B20 , 54E52

Keywords: asymmetric normed space , Baire category theorem , Hahn–Banach theorem , index of symmetry , topological classification

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 6 • December 2020
Back to Top