# Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces

## TL;DR

This paper investigates the conditions under which seminorms on infinite-dimensional vector spaces are continuous or discontinuous relative to various norm-topologies, clarifying their behavior in functional analysis.

## Contribution

It provides a detailed analysis of the continuity properties of seminorms on infinite-dimensional spaces, highlighting new criteria and characterizations.

## Key findings

- Characterization of continuous seminorms
- Conditions for discontinuity of seminorms
- Implications for functional analysis and topology

## Abstract

Let $S$ be a seminorm on an infinite-dimensional real or complex vector space $X$. Our purpose in this note is to study the continuity and discontinuity properties of $S$ with respect to certain norm-topologies on $X$.