Borel graph theorem

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] 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:[1]

Let X and Y be locally convex Hausdorff spaces and let ${\displaystyle u:X\to Y}$ 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 ${\displaystyle X\times Y}$, then u is continuous.

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space X is called a ${\displaystyle K_{\sigma \delta }}$ 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 ${\displaystyle K_{\sigma \delta }}$ space (that is, if there is a ${\displaystyle K_{\sigma \delta }}$ 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:[2]

Let X and Y be locally convex Hausdorff spaces and let ${\displaystyle u:X\to Y}$ 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 ${\displaystyle X\times Y}$, then u is continuous.