Corollary

— The topology $\mathscr{T}$ of a topological vector space E, is defined by a set of semi-norms (II, p. 3) if, and only if, $\mathscr{T}$ is locally convex.

The condition is necessary since every semi-norm on E is a convex function, and so, for $\alpha > 0$, the set of $x \in E$ for which $p(x) \leq \alpha$, is convex (II, p. 17, corollary). Conversely if V is a symmetric, closed, convex neighbourhood of 0 in E, the gauge $p$ of V is a semi-norm on E such that V is the set of points $x$ of E satisfying $p(x) \leq 1$ (II, p. 20, prop. 23).

This shows further that a locally convex topology $\mathscr{T}$ is defined by the set of all semi-norms that are continuous for $\mathscr{T}$. Further, if $\mathscr{T}$ is metrisable, then it is defined by an enumerable set of semi-norms.

From the corollary to prop. 1, all the results of § 1 on topologies defined by sets of semi-norms apply in particular to locally convex topologies over real vector spaces. A locally convex Hausdorff space E has a completion $\hat{E}$ that is locally convex. A complete, metrisable locally convex space is called a Fréchet space; every Banach space is a Fréchet space.