Hypocontinuous bilinear map
In mathematics, a hypocontinuous is a condition on bilinear maps of topological [vector space]s that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous.
Definition
If, and are topological vector spaces then a bilinear map is called hypocontinuous if the following two conditions hold:- for every bounded set the set of linear maps is an equicontinuous subset of, and
- for every bounded set the set of linear maps is an equicontinuous subset of.
Sufficient conditions
Theorem: Let X and Y be barreled spaces and let Z be a locally convex space. Then every separately continuous bilinear map of into Z is hypocontinuous.Examples
- If X is a Hausdorff locally convex barreled space over the field, then the bilinear map defined by is hypocontinuous.