Uniform Continuity - Generalization To Topological Vector Spaces

Generalization To Topological Vector Spaces

In the special case of two topological vector spaces and, the notion of uniform continuity of a map becomes: for any neighborhood of zero in, there exists a neighborhood of zero in such that implies

For linear transformations, 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.