Editing Uniform continuity (section)

== Generalization to topological vector spaces ==

In the special case of two [[topological vector spaces]] <math>V</math> and <math>W</math>, the notion of uniform continuity of a map <math>f:V\to W</math> becomes: for any neighborhood <math>B</math> of zero in <math>W</math>, there exists a neighborhood <math>A</math> of zero in <math>V</math> such that <math>v_1-v_2\in A</math> implies <math>f(v_1)-f(v_2)\in B.</math>

For [[linear transformation]]s <math>f:V\to W</math>, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in [[functional analysis]] to extend a linear map off a dense subspace of a [[Banach space]].