seminorm continuous at zero

Question

Let $X$ be a topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and $p$ be a seminorm on $X$. I suppose that $p$ is continuous at $0$ and I want to prove that $p$ is continuous.

The proof given here uses the sequential characterization of continuity : it takes a sequence $x_k\to x$ and proves that $p(x_k)\to p(x)$.

As far as I know, the sequential continuity implies the continuity only in the case of an Hausdorff metrisable space, which is not the case of a general topological vector space.

QUESTION 1: can one use the sequential characterization ? Why ? What do I miss ?

QUESTION 2 : If not, how do I prove the continuity of $p$ ?

Answer 1

You cannot use sequential continuity.

Let $\epsilon >0$. There is a neighborhood $U$ of $0$ such that $p(y)<\epsilon$ for $y \in U$. Replacing $U$ bu $U\cap (-U)$ we may assume that $U$ is symmetric. Let $x \in X$. $p(z)-p(x)\le p(z-x)<\epsilon$ whenever $z-x \in U$, hence whenvever $z \in x+U$. Similarly, $p(x)-p(z)<\epsilon$ whenever $z \in x-U=x+U$. Thus, $|p(x)-p(z)|<\epsilon$ whenever $z \in x-U=x+U$Since $X$ is a tvs, $x+U$ is a neighborhood of $x$. We have proved continuity at any point $x$

Answer 2

The proof given in the quoted course is in fact correct. The point is that it has to be understood in the sense of nets, not sequences.

So if $x_{\lambda}$ is a net converging to $x$, we have $(x_{\lambda}-x)\to 0$ and (very) few algebra shows that $p(x_{\lambda})\to p(x)$.

Now the convergence of net implies the continuity of the function $p$.