# Local compactness in right bounded asymmetric normed spaces

## TL;DR

This paper explores the properties of finite dimensional asymmetric normed spaces, focusing on right boundedness and its relation to compactness of the unit ball, revealing new insights and resolving open questions.

## Contribution

It characterizes finite dimensional right bounded asymmetric normed spaces and clarifies their compactness properties, including the existence of spaces with compact but not strongly compact unit balls.

## Key findings

- Not all right bounded asymmetric norms have compact closed balls.
- There exist finite dimensional spaces with compact but not strongly compact unit balls.
- A finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.

## Abstract

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. In the positive direction, we will prove that a finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.