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]

## Statement[edit]

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

## Generalization[edit]

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space  X {\displaystyle X} is called a  K σ δ {\displaystyle K_{\sigma \delta }} if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space  Y {\displaystyle Y} is called K-analytic if it is the continuous image of a  K σ δ {\displaystyle K_{\sigma \delta }} space (that is, if there is a  K σ δ {\displaystyle K_{\sigma \delta }} space  X {\displaystyle X} and a continuous map of  X {\displaystyle X} onto  Y {\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  X {\displaystyle X} and  Y {\displaystyle Y} be locally convex Hausdorff spaces and let  u : X -> Y {\displaystyle u:X\to Y} be linear. If  X {\displaystyle X} is the inductive limit of an arbitrary family of Banach spaces, if  Y {\displaystyle Y} is a K-analytic space, and if the graph of  u {\displaystyle u} is closed in  X × Y , {\displaystyle X\times Y,} then  u {\displaystyle u} is continuous.

## See also[edit]

* 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

## References[edit]

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

## Bibliography[edit]

* 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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