title: Hypocontinuous bilinear map
text: # 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 $X$, $Y$ and $Z$ are topological vector spaces then a bilinear map $\beta :X\times Y\to Z$ is called hypocontinuous if the following two conditions hold:
  
  * for every bounded set $A\subseteq X$ the set of linear maps $\{\beta (x,\cdot )\mid x\in A\}$ is an equicontinuous subset of $Hom(Y,Z)$, and
  * for every bounded set $B\subseteq Y$ the set of linear maps $\{\beta (\cdot ,y)\mid y\in B\}$ is an equicontinuous subset of $Hom(X,Z)$.
  
  ## 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 $X\times Y$ into Z is hypocontinuous.
  
  ## Examples
  
  * If X is a Hausdorff locally convex barreled space over the field $\mathbb {F}$, then the bilinear map $X\times X^{\prime }\to \mathbb {F}$ defined by $\left(x,x^{\prime }\right)\mapsto \left\langle x,x^{\prime }\right\rangle :=x^{\prime }\left(x\right)$ is hypocontinuous.
  