Closed graph theorem (functional analysis)

This article is about closed graph theorems in functional analysis. For other results with the same name, see Closed graph theorem.

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of the operator is closed (see also closed graph property).

The closed graph theorem has extensive application throughout functional analysis, because it can control whether a partially-defined linear operator admits continuous extensions. For this reason, it has been generalized to many circumstances beyond the elementary formulation above.

Explanation

Let $T:E\to F$ be a linear operator between Banach spaces (or more generally Fréchet spaces). Then the continuity of $T$ means that $Tx_{i}\to Tx$ for each convergent sequence $x_{i}\to x$. On the other hand, the closedness of the graph of $T$ means that for each convergent sequence $x_{i}\to x$ such that $Tx_{i}\to y$, then $y=Tx$. Hence, the closed graph theorem says that in order to check the continuity of $T$, one needs to show $Tx_{i}\to Tx$ under the additional assumption that $Tx_{i}$ is convergent.

Statement

Between Banach spaces

Closed Graph Theorem for Banach spaces—If $T:X\to Y$ is an everywhere-defined linear operator between Banach spaces, then the following are equivalent:

1. $T$ is continuous.
2. $T$ is closed (that is, the graph of $T$ is closed in the product topology on $X\times Y).$
3. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x$ in $X$ then $T\left(x_{\bullet }\right):=\left(T\left(x_{i}\right)\right)_{i=1}^{\infty }\to T(x)$ in $Y.$
4. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0$ in $X$ then $T\left(x_{\bullet }\right)\to 0$ in $Y.$
5. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x$ in $X$ and if $T\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y,$ then $y=T(x).$
6. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0$ in $X$ and if $T\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y,$ then $y=0.$

The operator is required to be everywhere-defined, that is, the domain $D(T)$ of $T$ is $X.$ This condition is necessary, as there exist closed linear operators that are unbounded (not continuous); a prototypical example is provided by the derivative operator on $C([0,1]),$ whose domain is a strict subset of $C([0,1]).$

The usual proof of the closed graph theorem employs the open mapping theorem. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the importance of $X$ and $Y$ being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.

Complete metrizable codomain

The closed graph theorem can be generalized from Banach spaces to more abstract topological vector spaces in the following ways.

Theorem—A linear operator from a barrelled space $X$ to a Fréchet space $Y$ is continuous if and only if its graph is closed.

Between F-spaces

There are versions that does not require $Y$ to be locally convex.

Theorem—A linear map between two F-spaces is continuous if and only if its graph is closed.

This theorem is restated and extend it with some conditions that can be used to determine if a graph is closed:

Theorem—If $T:X\to Y$ is a linear map between two F-spaces, then the following are equivalent:

1. $T$ is continuous.
2. $T$ has a closed graph.
3. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x$ in $X$ and if $T\left(x_{\bullet }\right):=\left(T\left(x_{i}\right)\right)_{i=1}^{\infty }$ converges in $Y$ to some $y\in Y,$ then $y=T(x).$
4. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0$ in $X$ and if $T\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y,$ then $y=0.$

Complete pseudometrizable codomain

Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.

Closed Graph Theorem—Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.

Closed Graph Theorem—A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous.

Codomain not complete or (pseudo) metrizable

Theorem—Suppose that $T:X\to Y$ is a linear map whose graph is closed. If $X$ is an inductive limit of Baire TVSs and $Y$ is a webbed space then $T$ is continuous.

Closed Graph Theorem—A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.

An even more general version of the closed graph theorem is

Theorem—Suppose that $X$ and $Y$ are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

If $G$ is any closed subspace of $X\times Y$ and $u$ is any continuous map of $G$ onto $X,$ then $u$ is an open mapping.

Under this condition, if $T:X\to Y$ is a linear map whose graph is closed then $T$ is continuous.

Borel graph theorem

Main article: Borel Graph Theorem

The Borel graph theorem, proved by L. Schwartz, 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:

Borel Graph Theorem—Let $u:X\to Y$ be linear map between two locally convex Hausdorff spaces $X$ and $Y.$ 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.

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 K-analytic 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 Frechet space. The generalized Borel graph theorem states:

Generalized Borel Graph Theorem—Let $u:X\to Y$ be a linear map between two locally convex Hausdorff spaces $X$ and $Y.$ 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.

Related results

If $F:X\to Y$ is closed linear operator from a Hausdorff locally convex TVS $X$ into a Hausdorff finite-dimensional TVS $Y$ then $F$ is continuous.

See also

• Almost open linear map – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Barrelled space – Type of topological vector space
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed linear operator
• Densely defined operator – Function that is defined almost everywhere (mathematics)
• Discontinuous linear map
• Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
• Webbed space – Space where open mapping and closed graph theorems hold

References

Notes

Bibliography

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