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 ,  and  are topological vector spaces then a bilinear map  is called hypocontinuous if the following two conditions hold:
 for every bounded set  the set of linear maps  is an equicontinuous subset of , and
 for every bounded set  the set of linear maps  is an equicontinuous subset of .

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  into Z is hypocontinuous.

Examples

 If X is a Hausdorff locally convex barreled space over the field , then the bilinear map  defined by  is hypocontinuous.

See also

References

Bibliography

 
  
  
  

Topological vector spaces
Bilinear maps