About locally convex Hausdorff topological vector space

Question

Let $E$ be a locally convex Hausdorff topological vector space. Show that $E$ is isomorphic to a subspace of a product of normed spaces.

All I know is that, if $E$ is locally convex Hausdorff, then there is a family of separated semi-norms. The problem at this point is, I have no idea how to construct(if necessary) such a product of normed space and what this space looks like. Furthermore, I think maybe this has something to do with the family of separated semi-norms, but I don't know how to connect the separated pieces.

Any hints are welcomed.

Answer 1

Furthermore, I think maybe this has something to do with the family of separated semi-norms

Very much so.

Let $E$ be a vector space (over $\mathbb{R}$ or $\mathbb{C}$ for simplicity), and $p \colon E \to [0,+\infty)$ a seminorm. Then

• $\ker p := \{ x \in E : p(x) = 0\}$ is a linear subspace of $E$, and

• $\hat{p} \colon E/\ker p \to [0,+\infty)$ given by $\hat{p}(x+\ker p) = p(x)$ is

  1. well-defined, and
  2. a norm on $E/\ker p$.

It should now not be hard to guess one appropriate product of normed spaces, and to verify all claims.