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.[1][2] 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.[3][4][5]

Statement of the theorem

It may be helpful to first recall the following terms:

  • A topological vector space 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 T1 space 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 convex set if, for any two points , the line segment joining them lies wholly within , i.e., 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 .)

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

Theorem. A topological vector space is normable if and only if it is a T1 space and admits a bounded convex neighbourhood of the origin.

See also

References

  1. 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.
  2. 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.
  3. Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics, No. 15. New York-Heidelberg: Springer-Verlag. ISBN 0387900802.
  4. Kolmogorov, A. N. (1934). "Zur Normierbarkeit eines allgemeinen topologischen linearen Räumes". Studia Math. 5.
  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)
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