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Borel graph theorem

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In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz.[1]

The Borel graph theorem shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis.[1]

Statement

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A topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space. The weak dual of a separable Fréchet space and the strong dual of a separable Fréchet–Montel space are Souslin spaces. Also, the space of distributions and all Lp-spaces over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states:[1]

Let and be Hausdorff locally convex spaces and let be linear. If is the inductive limit of an arbitrary family of Banach spaces, if is a Souslin space, and if the graph of is a Borel set in then is continuous.

Generalization

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An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space is called a if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space is called K-analytic if it is the continuous image of a space (that is, if there is a space and a continuous map of onto ). Every compact set is K-analytic so that there are non-separable K-analytic spaces. Also, every Polish, Souslin, and reflexive Fréchet space is K-analytic as is the weak dual of a Fréchet space. The generalized theorem states:[2]

Let and be locally convex Hausdorff spaces and let be linear. If is the inductive limit of an arbitrary family of Banach spaces, if is a K-analytic space, and if the graph of is closed in then is continuous.

See also

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References

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  1. ^ a b c Trèves 2006, p. 549.
  2. ^ Trèves 2006, pp. 557–558.

Bibliography

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  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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