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 ; that is, 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 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.
Statement of the theorem
Because translation 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".Definitions
It may be helpful to first recall the following terms:- A 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 if there is a norm on such that the open balls of the norm generate the given topology
- A topological space is called a 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 T1 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 T1.
- A subset of a vector space is a 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 if, for every open neighbourhood of the origin, there exists a scalar so that