# When topological vector space metrizable

This question comes from Chapter 5 of Folland's Real Analysis.It's a proposition (5.16)without proof.

Let $$X$$ be a vector space equipped with the topology defined by a family $$\left\{ p_{\alpha} \right\} _{\alpha \in A}$$ of seminorms.
a, $$X$$ is Hausdorff iff for each $$x\ne0$$ there exists $$\alpha \in A$$ such that $$p_\alpha(x)\ne 0$$.
b.If $$X$$ is Hausdorff and $$A$$ is countable,then $$X$$ is metrizable with a translation invariant metric (i.e.,$$\rho(x,y)=\rho(x+z,y+z)$$ for all $$x,y,z\in X$$).

The so-called topology defined by seminorms comes from the theorem:

Let $$\left\{ p_{\alpha} \right\} _{\alpha \in A}$$ be a family of seminorm on the vector space $$X$$.If $$x\in X,\alpha\in A$$,and $$\varepsilon>0$$,let $$U_{x\alpha \varepsilon}=\left\{ y\in X:p_{\alpha}\left( y-x \right) <\varepsilon \right\} ,$$ and let $$\mathcal{T}$$ be the topology generated by the sets $$U_{x\alpha\varepsilon}$$.Then $$(X,\mathcal{T})$$ is a locally convex topological vector space.

I have solved the problem $$a$$,but I cannot make any progress on b.

We consider the seminorms $$\{p_\alpha\}_{\alpha\in A}$$ which define the topology on your vector space. In part (b), we assume that $$X$$ is a Hausdorff space and that $$A$$ is countable. Thus we may as well assume that $$A=\mathbb{N}$$. Now I claim that the function $$\rho:X\times X\to [0,\infty)$$ given by $$\rho(x,y) = \sum_{k=1}^\infty \frac{1}{2^k} \frac{p_k(x-y)}{1+p_k(x-y)}$$ is a metric which is translation invariant.
It is clear from the formulae and the definition of a seminorm that $$\rho$$ is translation invariant and symmetric. I will leave it to you to prove that $$\rho$$ is finite valued, positive definite, and satisfies the triangle inequality. (In checking that $$\rho$$ is positive definite, you will need the assumption that $$X$$ is Hausdorff and the result of part (a).)