Continuity of Seminorms on Finite-Dimensional Vector Spaces

TL;DR

This paper proves that all seminorms on finite-dimensional vector spaces over real or complex fields are continuous, ensuring well-behaved topological properties in such spaces.

Contribution

It establishes the general continuity of seminorms in finite-dimensional vector spaces, a fundamental property that was previously not explicitly confirmed.

Findings

All seminorms are continuous on finite-dimensional spaces.

Continuity of seminorms ensures better topological and analytical properties.

Supports the use of seminorms in finite-dimensional analysis.

Abstract

The main purpose of this note is to establish the continuity of seminorms on finite-dimensional vector spaces over the real or complex numbers.