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

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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

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Theorem:[1] 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

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  • If X is a Hausdorff locally convex barreled space over the field  , then the bilinear map   defined by   is hypocontinuous.[1]

See also

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References

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  1. ^ a b Trèves 2006, pp. 424–426.

Bibliography

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  • 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) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.