Asymmetric normed Baire space

TL;DR

This paper investigates the conditions under which asymmetric normed spaces are Baire spaces, establishing that only those isomorphic to their associated normed spaces possess the Baire property.

Contribution

It provides a characterization of biBanach asymmetric normed spaces that are Baire spaces, linking the Baire property to isomorphism with associated normed spaces.

Findings

A biBanach asymmetric normed space is a Baire space iff it is isomorphic to its associated normed space.

Asymmetric normed spaces are generally not Baire spaces unless equivalent to normed spaces.

The topology induced by an asymmetric norm must be equivalent to a norm topology for the space to be Baire.

Abstract

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.