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[]

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[]

It may be helpful to first recall the following terms:

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

See also[]

• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Normed space
• Topological vector space – Vector space with a notion of nearness

References[]