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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 T_{1} 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".

It may be helpful to first recall the following terms:

- A
*topological vector space*(TVS) is a vector space equipped with a topology such that the vector space operations of scalar multiplication and vector addition are continuous. - A topological vector space is called
*normable*if there is a norm on such that the open balls of the norm generate the given topology (Note well that a given normable topological vector space might admit multiple such norms.) - A topological space is called a
*T*if, for every two distinct points there is an open neighbourhood of that does not contain In a topological vector space, this is equivalent to requiring that, for every there is an open neighbourhood of the origin not containing Note that being T_{1}space_{1}is weaker than being a Hausdorff space, in which every two distinct points admit open neighbourhoods of and of with ; since normed and normable spaces are always Hausdorff, it is a "surprise" that the theorem only requires T_{1}. - A subset of a vector space is a
*convex set*if, for any two points the line segment joining them lies wholly within that is, for all - A subset of a topological vector space is a
*bounded set*if, for every open neighbourhood of the origin, there exists a scalar so that (One can think of as being "small" and as being "big enough" to inflate to cover )

- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Normed vector space – Vector space on which a distance is defined
- Topological vector space – Vector space with a notion of nearness

**^**Papageorgiou, Nikolaos S.; Winkert, Patrick (2018).*Applied Nonlinear Functional Analysis: An Introduction*. Walter de Gruyter. Theorem 3.1.41 (Kolmogorov's Normability Criterion). ISBN 9783110531831.**^**Edwards, R. E. (2012). "Section 1.10.7: Kolmagorov's Normability Criterion".*Functional Analysis: Theory and Applications*. Dover Books on Mathematics. Courier Corporation. pp. 85–86. ISBN 9780486145105.**^**Berberian, Sterling K. (1974).*Lectures in Functional Analysis and Operator Theory*. Graduate Texts in Mathematics, No. 15. New York-Heidelberg: Springer-Verlag. ISBN 0387900802.**^**Kolmogorov, A. N. (1934). "Zur Normierbarkeit eines allgemeinen topologischen linearen Räumes".*Studia Math*.**5**.**^**Tikhomirov, Vladimir M. (2007). "Geometry and approximation theory in A. N. Kolmogorov's works". In Charpentier, Éric; Lesne, Annick; Nikolski, Nikolaï K. (eds.).*Kolmogorov's Heritage in Mathematics*. Berlin: Springer. pp. 151–176. doi:10.1007/978-3-540-36351-4_8. (See Section 8.1.3)