Hypocontinuous bilinear map

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 ${\displaystyle Hom(X,Z)}$.

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.

• 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]

• bilinear map

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13627-9
• Schaefer, Helmut H; Wolff, M.P. (1999), Topological vector spaces (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98726-2
• Trèves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)