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# Index of symmetry and topological classification of asymmetric normed spaces.

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

https://hal-paris1.archives-ouvertes.fr/hal-02864982
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Submitted on : Thursday, June 11, 2020 - 2:31:45 PM
Last modification on : Friday, May 6, 2022 - 4:52:02 PM

### Identifiers

• HAL Id : hal-02864982, version 1

### Citation

Mohammed Bachir, Gonzalo Flores. Index of symmetry and topological classification of asymmetric normed spaces.. Rocky Mountain J. Math., 2020, 50 (6), pp.1951-1964. ⟨hal-02864982⟩

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