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; 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]

Statement of the theorem

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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".

Definitions

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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 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   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  )

See also

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References

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  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)