Borel graph theorem

In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz.Template:Sfn

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.Template:Sfn

Statement

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:Template:Sfn

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

Generalization

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space $X$ is called a $K_{\sigma \delta }$ if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space $Y$ is called Template:Visible anchor if it is the continuous image of a $K_{\sigma \delta }$ space (that is, if there is a $K_{\sigma \delta }$ 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 Fréchet space is K-analytic as is the weak dual of a Fréchet space. The generalized theorem states:Template:Sfn

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

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link

Bibliography

• Template:Trèves François Topological vector spaces, distributions and kernels