Proposition 2

On a vector space $E$, let $\mathscr{T}$ be the topology defined by a set of semi-norms $\Gamma$.

(i) The closure of {0} in E, for $\mathscr{T}$, is the subset of $x \in E$ for which $p(x) = 0$ for every semi-norm $p \in \Gamma$.

(ii) If $\mathscr{T}$ is Hausdorff and $\Gamma$ is enumerable, then $\mathscr{T}$ is metrisable.

The proposition follows immediately from the definitions and from GT, IX, § 2 . 4, cor. 1.

Note that if $\mathscr{T}$ is metrisable, it may be that $\mathscr{T}$ cannot be defined by a single norm; this is the case in the example given above (cf. IV, p. 18, Example 4).

Let E be a vector space over K, with the topology defined by a set of semi-norms $\Gamma$. Let $\hat{E}$ be the Hausdorff completion of E (I, p. 6), and $\hat{\Gamma}$ be the set of mappings $\hat{p}$ of $\hat{E}$ in $\mathbf{R}_+$ where $p$ varies in $\Gamma$ (GT, II, § 3 . 7, prop. 15). By the principle of extending inequalities, the functions $\hat{p} \in \hat{\Gamma}$ are semi-norms on $\hat{E}$, and the functions $\hat{p}(x - y)$ form a set of pseudometrics defining the uniform structure of $\hat{E}$ (GT, IX, § 1 . 3, prop. 1). We see, therefore, that $\hat{\Gamma}$ is a fundamental set of semi-norms defining the topology of $\hat{E}$.