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

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

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 [math]\displaystyle{ X \times Y }[/math] into Z is hypocontinuous.

Examples

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

See also

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

Bibliography

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13627-9 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.