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:Template:Sfn 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 $𝔽$, then the bilinear map $X\times X^{\prime }\to 𝔽$ defined by $(x,x^{\prime })\mapsto ⟨x,x^{\prime }⟩:=x^{\prime }\left(x\right)$ is hypocontinuous.Template:Sfn

See also

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Bibliography

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• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels