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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.
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https://hal-paris1.archives-ouvertes.fr/hal-02488691
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Submitted on : Thursday, June 4, 2020 - 3:44:02 AM
Last modification on : Friday, May 6, 2022 - 4:50:07 PM

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• HAL Id : hal-02488691, version 2
• ARXIV : 2002.10166

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Mohammed Bachir, G. Flores. INDEX OF SYMMETRY AND TOPOLOGICAL CLASSIFICATION OF ASYMMETRIC NORMED SPACES. 2020. ⟨hal-02488691v2⟩

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