Kolmogorov's normability criterion

In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be normable, i.e. for the existence of a norm on the space that generates the given topology. The normability criterion can be seen as a result in same vein as the Nagata–Smirnov metrization theorem, which gives a necessary and sufficient condition for a topological space to be metrizable. The result was proved by the Russian mathematician Andrey Nikolayevich Kolmogorov in 1934.

Statement of the theorem

It may be helpful to first recall the following terms:

• A topological vector space is a vector space $X$ equipped with a topology $\mathcal{T}$ such that the vector space operations of scalar multiplication and vector addition are continuous.
• A topological topological vector space $(X, \mathcal{T})$ is called normable if there is a norm $\| \cdot \| \colon X \to \mathbb{R}$ on $X$ such that the open balls of the norm $\| \cdot \|$ generate the given topology $\mathcal{T}$. (Note well that a given normable topological vector space might admit multiple such norms.)
• A topological space $(X, \mathcal{T})$ is called a T₁ space if, for every two distinct points $x, y \in X$, there is an open neighbourhood $U_{x}$ of $x$ that does not contain $y$. In a topological vector space, this is equivalent to requiring that, for every $x \neq 0$, there is an open neighbourhood of the origin not containing $x$. Note that being T₁ is weaker than being a Hausdorff space, in which every two distinct points $x, y \in X$ admit open neighbourhoods $U_{x}$ of $x$ and $U_{y}$ of $y$ with $U_{x} \cap U_{y} = \varnothing$; since normed and normable spaces are always Hausdorff, it is a “surprise” that the theorem only requires T₁.
• A subset $A$ of a vector space $X$ is a convex set if, for any two points $x, y \in A$, the line segment joining them lies wholly within $A$, i.e., for all $0 \leq t \leq 1$, $(1 - t) x + t y \in A$.
• A subset $A$ of a topological vector space $(X, \mathcal{T})$ is a bounded set if, for every open neighbourhood $U$ of the origin, there exists a scalar $\lambda$ so that $A \subseteq \lambda U$. (One can think of $U$ as being “small” and $\lambda$ as being “big enough” to inflate $U$ to cover $A$.)

Expressed in these terms, Kolmogorov's normability criterion is as follows:

Theorem. A topological vector space $(X, \mathcal{T})$ is normable if and only if it is a T₁ space and admits a bounded convex neighbourhood of the origin.