Locally Convex Topological Vector Space - Continuous Linear Mappings

Continuous Linear Mappings

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces V and W with families of seminorms {p_α}_α and {q_β}_β respectively, a linear map T from V to W is continuous if and only if for every β there exist α₁, α₂, ..., α_n and exists an M>0 such that for all v in V

In other words, each seminorm of the range of T is bounded above by some finite sum of seminorms in the domain. If the family {p_α}_α is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.