Hypocontinuous bilinear map

(Redirected from Hypocontinuity)

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

References

^ Trèves 2006, pp. 424–426.

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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

v t e Topological tensor products and nuclear spaces
Basic concepts	Auxiliary normed spaces Nuclear space Tensor product Topological tensor product of Hilbert spaces
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators/Maps	Fredholm determinant Fredholm kernel Hilbert–Schmidt operator Hypocontinuity Integral Nuclear between Banach spaces Trace class
Theorems	Grothendieck trace theorem Schwartz kernel theorem

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic / Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

Categories:

Misplaced Pages