Continuity and discontinuity of seminorms on infinite-dimensional vector spaces (Q2321353)

Let \(X\) be a vector space over a field \(\mathbb{F}\) (either \(\mathbb{R}\) or \(\mathbb{C}\)). If \(X\) is finite-dimensional, then all norms on \(X\) are equivalent and hence induce a unique norm topology on \(X\). If \(X\) is infinite-dimensional, then not all norms on \(X\) are equivalent and consequently the induced norm topology is not unique. The authors prove that, if \(X\) is infinite-dimensional and if \(S\ne 0\) is a seminorm on \(X\), then there exists a norm on \(X\) with respect to which \(S\) is ubiquitously continuous and also there exists a norm on \(X\) with respect to which \(S\) is ubiquitously discontinuous. As a corollary, the authors deduce that \(S\) is continuous with respect to every norm on \(X\) if and only if \(X\) is finite-dimensional.