Hypocontinuous bilinear map

In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous. 
Definition.
If formula_1, formula_2 and formula_3 are topological vector spaces then a bilinear map formula_4 is called "hypocontinuous" if the following two conditions hold:
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 formula_11 into "Z" is hypocontinuous.