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 that function is closed.
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.
The closed graph theorem is a result about linear map between two vector spaces endowed with topologies making them into topological vector spaces (TVSs). We will henceforth assume that and are topological vector spaces, such as Banach spaces for example, and that Cartesian products, such as are endowed with the product topology. The graph of this function is the subset
A closed linear operator is a linear map whose graph is closed (it need not be continuous or bounded). It is common in functional analysis to call such maps "closed", but this should not be confused the non-equivalent notion of a "closed map" that appears in general topology.
Partial functions
It is common in functional analysis to consider partial functions, which are functions defined on a dense subset of some space A partial function is declared with the notation which indicates that has prototype (that is, its domain is and its codomain is ) and that is a dense subset of Since the domain is denoted by it is not always necessary to assign a symbol (such as ) to a partial function's domain, in which case the notation or may be used to indicate that is a partial function with codomain whose domain is a dense subset of ^{[1]} A densely defined linear operator between vector spaces is a partial function whose domain is a dense vector subspace of a TVS such that is a linear map. A prototypical example of a partial function is the derivative operator, which is only defined on the space of once continuously differentiable functions, a dense subset of the space of continuous functions.
Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function is (as before) the set However, one exception to this is the definition of "closed graph". A partial function is said to have a closed graph (respectively, a sequentially closed graph) if is a closed (respectively, sequentially closed) subset of in the product topology; importantly, note that the product space is and not as it was defined above for ordinary functions.^{[note 1]}
A linear operator is closable in if there exists a vector subspace containing and a function (resp. multifunction) whose graph is equal to the closure of the set in Such an is called a closure of in , is denoted by and necessarily extends
If is a closable linear operator then a core or an essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).
Throughout, let and be topological spaces and is endowed with the product topology.
Main article: Closed graph property |
If is a function then it is said to have a closed graph if it satisfies any of the following are equivalent conditions:
and if is a Hausdorff compact space then we may add to this list:
and if both and are first-countable spaces then we may add to this list:
Function with a sequentially closed graph
If is a function then the following are equivalent:
Suppose is a linear operator between Banach spaces.
Closed Graph Theorem for Banach spaces — If is an everywhere-defined linear operator between Banach spaces, then the following are equivalent:
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 (not continuous); 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 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 to a Fréchet space is continuous if and only if its graph is closed.
There are versions that does not require to be locally convex.
Theorem — A linear map between two F-spaces is continuous if and only if its graph is closed.^{[7]}^{[8]}
This theorem is restated and extend it with some conditions that can be used to determine if a graph is closed:
Theorem — If is a linear map between two F-spaces, then the following are equivalent:
Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.
Closed Graph Theorem^{[10]} — 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.^{[10]}
Theorem^{[11]} — Suppose that is a linear map whose graph is closed. If is an inductive limit of Baire TVSs and is a webbed space then is continuous.
Closed Graph Theorem^{[10]} — 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^{[12]} — Suppose that and are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:
Under this condition, if is a linear map whose graph is closed then is continuous.
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.^{[13]} 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 be linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a Souslin space, and if the graph of is a Borel set in then is continuous.^{[13]}
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 (that is, if there is a space and a continuous map of onto ).
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^{[14]} — Let be a linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a K-analytic space, and if the graph of is closed in then is continuous.
If is closed linear operator from a Hausdorff locally convex TVS into a Hausdorff finite-dimensional TVS then is continuous.^{[15]}