Proposition 1

Let E be a left topological vector space over K and let p be a semi-norm on E. The following conditions are equivalent : a) p is continuous in E. b) p is continuous at the point 0. c) p is uniformly continuous. d) For each real number $\alpha > 0$, the set W(p, $\alpha$), of those $x \in E$ for which $p(x) < \alpha$, is open in E. e) There exists a real number $\alpha > 0$, such that W(p, $\alpha$) is a neighbourhood of 0 in E. f) For every real number $\alpha > 0$, the set V(p, $\alpha$), of those $x \in E$ for which $p(x) \leq \alpha$, is a neighbourhood of 0 in E.

In fact, the implications c) $\Rightarrow$ a) $\Rightarrow$ b) $\Rightarrow$ d) $\Rightarrow$ e) $\Rightarrow$ f) $\Rightarrow$ c) follow immediately from (SN$_I$) and inequality (1).