Every seminorm on a finite dimensional Hausdorff TVS is continuous.

Question

Let $E$ be a topological vector space Hausdorff and $p:E \longrightarrow \mathbb{R}_+$ any seminorm on $E$. If $E$ is finite dimensional, say $dim(E)=n$, then I want to prove that $p$ is continuous. I thought about proving that the set $$B:=\{ x \in E \; ; \; p(x)<1\}$$ is a open set. But I don't know how to relate this to the fact that $dim(E)=n$.

Answer 1

It is well known that the topology of a finite dimensional Hausdorff vector space is induced by a norm, so we assume $E$ to be a normed space.

Let $N=\{ x\in E\mid p(x)=0\}$, this is a vector sub-space as can be easily verified. Now $p$ descends to $E/N$, where it is a norm and not just a semi-norm, equip $E/N$ with this norm. The projection: $$\pi: E\to E/N$$ is a linear map between finite dimensional normed spaces. As such it is continuous. Post-compose with $p$ to get that the map: $$x\mapsto \pi(x)\mapsto p(\pi(x))=p(x)$$ is continuous. As such $p$ is continuous on $E$.