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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.

Preliminaries

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

of where denotes the function's domain. The map is said to have a closed graph (in ) if its graph is a closed subset of product space (with the usual product topology). Similarly, is said to have a sequentially closed graph if is a sequentially closed subset of

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]}

Closable maps and closures

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 ).

Characterizations of closed graphs (general topology)

Throughout, let and be topological spaces and is endowed with the product topology.

Function with a closed graph

If is a function then it is said to have a closed graph if it satisfies any of the following are equivalent conditions:

1. (Definition): The graph of is a closed subset of
2. For every and net in such that in if is such that the net in then ^[2]
   • Compare this to the definition of continuity in terms of nets, which recall is the following: for every and net in such that in in
   • Thus to show that the function has a closed graph, it may be assumed that converges in to some (and then show that ) while to show that is continuous, it may not be assumed that converges in to some and instead, it must be proven that this is true (and moreover, it must more specifically be proven that converges to in ).

and if is a Hausdorff compact space then we may add to this list:

3. is continuous.^[3]

and if both and are first-countable spaces then we may add to this list:

4. has a sequentially closed graph in

Function with a sequentially closed graph

If is a function then the following are equivalent:

1. has a sequentially closed graph in
2. Definition: the graph of is a sequentially closed subset of
3. For every and sequence in such that in if is such that the net in then ^[2]

Basic properties of maps with closed graphs

Suppose is a linear operator between Banach spaces.

• If is closed then is closed where is a scalar and is the identity function.
• If is closed, then its kernel (or nullspace) is a closed vector subspace of
• If is closed and injective then its inverse is also closed.
• A linear operator admits a closure if and only if for every and every pair of sequences and in both converging to in such that both and converge in one has

Examples and counterexamples

Continuous but not closed maps

• Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology (where is not Hausdorff and that every function valued in is continuous). Let be defined by and for all Then is continuous but its graph is not closed in ^[2]
• If is any space then the identity map is continuous but its graph, which is the diagonal is closed in if and only if is Hausdorff.^[4] In particular, if is not Hausdorff then is continuous but not closed.
• If is a continuous map whose graph is not closed then is not a Hausdorff space.

Closed but not continuous maps

• If is a Hausdorff TVS and is a vector topology on that is strictly finer than then the identity map a closed discontinuous linear operator.^[5]
• Consider the derivative operator where is the Banach space of all continuous functions on an interval If one takes its domain to be then is a closed operator, which is not bounded.^[6] On the other hand, if is the space of smooth functions scalar valued functions then will no longer be closed, but it will be closable, with the closure being its extension defined on
• Let and both denote the real numbers with the usual Euclidean topology. Let be defined by and for all Then has a closed graph (and a sequentially closed graph) in but it is not continuous (since it has a discontinuity at ).^[2]
• Let denote the real numbers with the usual Euclidean topology, let denote with the discrete topology, and let be the identity map (i.e. for every ). Then is a linear map whose graph is closed in but it is clearly not continuous (since singleton sets are open in but not in ).^[2]

Closed graph theorems

Between Banach spaces

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.

Complete metrizable codomain

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

Between F-spaces

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

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

Complete pseudometrizable codomain

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

Codomain not complete or (pseudo) metrizable

An even more general version of the closed graph theorem is

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:

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:

Related results

If is closed linear operator from a Hausdorff locally convex TVS into a Hausdorff finite-dimensional TVS then is continuous.^[15]

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 – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• 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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