# Borel Graph Theorem

### (Functional Analysis)

Tech

###### Functional Analysis ## What is Borel graph theorem?

In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz. 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. Recall that 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: Let X and Y be locally convex Hausdorff spaces and let be linear. If X is the inductive limit of an arbitrary family of Banach spaces, if Y is a Souslin space, and if the graph of u is a Borel set in , then u is continuous. An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space X is called a if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space Y is called K-analytic if it is the continuous image of a space (that is, if there is a space X and a continuous map of X onto Y). Every compact set is K-analytic so that there are non-separable K-analytic spaces.Also, every Polish, Souslin, and reflexive Frechet space is K-analytic as is the weak dual of a Frechet space. The generalized theorem states: Let X and Y be locally convex Hausdorff spaces and let be linear. If X is the inductive limit of an arbitrary family of Banach spaces, if Y is a K-analytic space, and if the graph of u is closed in , then u is continuous.

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functional analysis

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— Date merged: 11/6/2021, 1:32:49 PM
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