Kolmogorov's normability criterion

Statement of the theorem

• 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.
• 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.