In a finite-dimentional Hausdorff locally convex vector space, how to prove there exists a seminorm which is a norm?

Let E be a finite-dimensional Hausdorff locally convex vector space, and $e_1,\ldots, e_n$ is its basis. I know that from the Hausdorff and locally convexity, there exists a seminorm $p$ satisfying that $p(e_1)\geq 1,\ldots$, and $p(e_n)\geq 1$. I want to prove that for any $x\neq 0$, $p(x)>0$. This immediately means that $p$ is a continous norm on this tvs.

By the way, I know that a finite-dimensional locally convex vector space can be induced by a norm. This has been discussed in this post. @Stephen Montgomery-Smith has posted a detailed solution. But I really want to prove that:

If $p(e_1)\geq 1,\ldots$, and $p(e_n)\geq 1$, then for any $x\neq 0$, $p(x)>0$.

In fact, if the above is true, then the fact that a finite-dimensional locally convex vector space can be induced by a norm would be proved more directly!

Solution 1:

Define the functional $$\omega: E \to \mathbb{C}: \sum_i \alpha_i e_i \mapsto \sum_i \alpha_i.$$

Then $p(x) = |\omega(x)|$ is a seminorm. Note that $$p(e_i) = 1$$ for all $i$ but we have $p(e_1-e_2) = |\omega(e_1-e_2)| = 0$ so your claim does not hold.

However, proving that $E$ admits a norm is quite easy. For example, define $$\left\|\sum_i \alpha_i e_i\right\|:= \sum_i |\alpha_i|.$$