In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In mathematics, there are several results known as the "closed graph theorem".
If is a map between topological spaces then the graph of is the set or equivalently, We say that the graph of is closed if is a closed subset of . Any continuous function into a Hausdorff space has a closed graph. However, if is a non-Hausdorff space then the graph of the identity map is the diagonal, which is closed in if and only if is Hausdorff. A function from a compact Hausdorff space into a Hausdorff space is continuous if and only if it has a closed graph.
In point-set topology, the closed graph theorem states the following: If is a topological space and is a compact Hausdorff space, then the graph of is closed if and only if is continuous.
For set-valued functions
The closed graph theorem for set-valued functions says that, for a compact Hausdorff range space, a set-valued function has a closed graph if and only if it is upper hemicontinuous and is a closed set for all.
Let be a Hausdorff TVS and let be a vector topology on that is strictly finer than. Then the identity map a closed discontinuous linear operator.
Closed graph theorems
The closed graph theorem can be generalized from Banach spaces to more abstract topological vector spaces in the following ways: and there are versions that does not require to be locally convex: We restate this theorem and extend it with some conditions that can be used to determine if a graph is closed:
is continuous;
has a closed graph;
If in and if converges in to some, then ;
If in and if converges in to some, then.
An even more general version of the closed graph theorem is
Between Banach spaces
In functional analysis, the closed graph theorem states the following: If and are Banach spaces, and is a linear operator, then is continuous if and only if its graph is closed in . The closed graph theorem can be reformulated may be rewritten into a form that is more easily usable: If is a linear operator between Banach spaces, then the following are equivalent:
Note that the operator is required to be everywhere-defined, that is, the domain of is. This condition is necessary, as there exist closed linear operators that are unbounded ; a prototypical example is provided by the derivative operator on, whose domain is a strict subset of. 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 and 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.
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: An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space is called a if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space is called K-analytic if it is the continuous image of a space. Every compact set is K-analytic so that there are non-separable K-analytic spaces. Also, every Polish, Souslin, and reflexiveFrechet space is K-analytic as is the weak dual of a Frechet space. The generalized theorem states:
