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 <math>X</math>, <math>Y</math> and <math>Z</math> are topological vector spaces then a bilinear map <math>\beta: X\times Y\to Z</math> is called ''hypocontinuous'' if the following two conditions hold: * for every bounded set <math>A\subseteq X</math> the set of linear maps <math>\{\beta(x,\cdot) \mid x\in A\}</math> is an equicontinuous subset of <math>Hom(Y,Z)</math>, and * for every bounded set <math>B\subseteq Y</math> the set of linear maps <math>\{\beta(\cdot,y) \mid y\in B\}</math> is an equicontinuous subset of <math>Hom(X,Z)</math>.
==Sufficient conditions==
'''Theorem''':{{sfn | Trèves | 2006 | pp=424-426}} Let ''X'' and ''Y'' be barreled spaces and let ''Z'' be a locally convex space. Then every separately continuous bilinear map of <math>X \times Y</math> into ''Z'' is hypocontinuous.
==Examples==
* If ''X'' is a Hausdorff locally convex barreled space over the field <math>\mathbb{F}</math>, then the bilinear map <math>X \times X^{\prime} \to \mathbb{F}</math> defined by <math>\left( x, x^{\prime} \right) \mapsto \left\langle x, x^{\prime} \right\rangle := x^{\prime}\left( x \right)</math> is hypocontinuous.{{sfn | Trèves | 2006 | pp=424-426}}
==See also==
* {{annotated link|Bilinear map}} * {{annotated link|Dual system}}
==References==
{{reflist}}
==Bibliography==
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Topological vector spaces | publisher=Springer-Verlag | location=Berlin, New York | series=Elements of mathematics | isbn=978-3-540-13627-9 | year=1987}} * {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} --> * {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} --> * {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->
{{Topological tensor products and nuclear spaces}} {{Functional analysis}}
Category:Topological vector spaces Category:Bilinear maps