Hypocontinuous bilinear map explained

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 x Y\toZ

is called hypocontinuous if the following two conditions hold:

A\subseteqX

the set of linear maps

\{\beta(x,)\midx\inA\}

is an equicontinuous subset of

Hom(Y,Z)

, and

B\subseteqY

the set of linear maps

\{\beta(,y)\midy\inB\}

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 x Y

into Z is hypocontinuous.

Examples

F

, then the bilinear map

X x X\prime\toF

defined by

\left(x,x\prime\right)\mapsto\left\langlex,x\prime\right\rangle:=x\prime\left(x\right)

is hypocontinuous.