In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be normable; that is, for the existence of a norm on the space that generates the given topology.
[1][2] The normability criterion can be seen as a result in same vein as the Nagata–Smirnov metrization theorem and Bing 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.
[3][4][5] Kolmogorov's normability criterion — A topological vector space is normable if and only if it is a T1 space and admits a bounded convex neighbourhood of the origin.
Because translation (that is, vector addition) by a constant preserves the convexity, boundedness, and openness of sets, the words "of the origin" can be replaced with "of some point" or even with "of every point".