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]

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 ${\displaystyle X}$ and ${\displaystyle Y}$ be Hausdorff locally convex spaces and let ${\displaystyle u:X\to Y}$ be linear. If ${\displaystyle X}$ is the inductive limit of an arbitrary family of Banach spaces, if ${\displaystyle Y}$ is a Souslin space, and if the graph of ${\displaystyle u}$ is a Borel set in ${\displaystyle X\times Y,}$ then ${\displaystyle u}$ is continuous.

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space ${\displaystyle X}$ is called a ${\displaystyle K_{\sigma \delta }}$ if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space ${\displaystyle Y}$ 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 ${\displaystyle X}$ and a continuous map of ${\displaystyle X}$ onto ${\displaystyle Y}$). 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:^[2]

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally convex Hausdorff spaces and let ${\displaystyle u:X\to Y}$ be linear. If ${\displaystyle X}$ is the inductive limit of an arbitrary family of Banach spaces, if ${\displaystyle Y}$ is a K-analytic space, and if the graph of ${\displaystyle u}$ is closed in ${\displaystyle X\times Y,}$ then ${\displaystyle u}$ is continuous.

• Closed graph property – Graph of a map closed in the product space
• Closed graph theorem – Theorem relating continuity to graphs
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Graph of a function – Representation of a mathematical function

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