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. 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.^[1] ^[2] ^[3]

Statement of the theorem

It may be helpful to first recall the following terms:

A topological vector space is a vector space $X$ equipped with a topology ${\mathcal {T}}$ such that the vector space operations of scalar multiplication and vector addition are continuous.
A topological vector space $(X,{\mathcal {T}})$ is called normable if there is a norm $\|\cdot \|\colon X\to \mathbb {R}$ on $X$ such that the open balls of the norm $\|\cdot \|$ generate the given topology ${\mathcal {T}}$ . (Note well that a given normable topological vector space might admit multiple such norms.)
A topological space $(X,{\mathcal {T}})$ is called a T₁ space if, for every two distinct points $x, y \in X$ , there is an open neighbourhood $U_{x}$ of $x$ that does not contain $y$ . In a topological vector space, this is equivalent to requiring that, for every $x\neq 0$ , there is an open neighbourhood of the origin not containing $x$ . Note that being T₁ is weaker than being a Hausdorff space, in which every two distinct points $x, y \in X$ admit open neighbourhoods $U_{x}$ of $x$ and $U_{y}$ of $y$ with $U_{x}\cap U_{y}=\varnothing$ ; since normed and normable spaces are always Hausdorff, it is a “surprise” that the theorem only requires T₁.
A subset $A$ of a vector space $X$ is a convex set if, for any two points $x,y\in A$ , the line segment joining them lies wholly within $A$ , i.e., for all $0\leq t\leq 1$ , $(1-t)x+ty\in A$ .
A subset $A$ of a topological vector space $(X,{\mathcal {T}})$ is a bounded set if, for every open neighbourhood $U$ of the origin, there exists a scalar $\lambda$ so that $A\subseteq \lambda U$ . (One can think of $U$ as being “small” and $\lambda$ as being “big enough” to inflate $U$ to cover $A$ .)

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

Theorem. A topological vector space $(X,{\mathcal {T}})$ is normable if and only if it is a T₁ space and admits a bounded convex neighbourhood of the origin.

References

↑ 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. Kolmogorov's Heritage in Mathematics. Berlin: Springer. pp. 151&ndash, 176. doi:10.1007/978-3-540-36351-4_8. (See Section 8.1.3)

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[Berberian1974-1] Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics, No. 15. New York-Heidelberg: Springer-Verlag. ISBN 0387900802.

[Kolmogorov1934-2] Kolmogorov, A. N. (1934). "Zur Normierbarkeit eines allgemeinen topologischen linearen Räumes". Studia Math. 5.

[Tikhomirov2007-3] Tikhomirov, Vladimir M. (2007). "Geometry and approximation theory in A. N. Kolmogorov's works". In Charpentier, Éric; Lesne, Annick; Nikolski, Nikolaï K. Kolmogorov's Heritage in Mathematics. Berlin: Springer. pp. 151&ndash, 176. doi:10.1007/978-3-540-36351-4_8. (See Section 8.1.3)