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 ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$ are topological vector spaces then a bilinear map ${\displaystyle \beta :X\times Y\to Z}$ is called hypocontinuous if the following two conditions hold:

• for every bounded set ${\displaystyle A\subseteq X}$ the set of linear maps ${\displaystyle \{\beta (x,\cdot )\mid x\in A\}}$ is an equicontinuous subset of ${\displaystyle Hom(Y,Z)}$, and
• for every bounded set ${\displaystyle B\subseteq Y}$ the set of linear maps ${\displaystyle \{\beta (\cdot ,y)\mid y\in B\}}$ is an equicontinuous subset of ${\displaystyle Hom(X,Z)}$.

Sufficient conditions

Theorem:^[1] Let X and Y be barreled spaces and let Z be a locally convex space. Then every separately continuous bilinear map of ${\displaystyle X\times Y}$ into Z is hypocontinuous.

Examples

• If X is a Hausdorff locally convex barreled space over the field ${\displaystyle \mathbb {F} }$, then the bilinear map ${\displaystyle X\times X^{\prime }\to \mathbb {F} }$ defined by ${\displaystyle \left(x,x^{\prime }\right)\mapsto \left\langle x,x^{\prime }\right\rangle$ :=x^{\prime }\left(x\right)} is hypocontinuous.^[1]

See also

• Bilinear map – Function of two vectors linear in each argument
• Dual system