View source for Closed graph theorem

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| footer    = The graph of the [[cubic function|cubic]] function {{nowrap|1=''f''(''x'') = ''x''<sup>3</sup> − 9''x''}} on the interval <nowiki>[-4,4]</nowiki> is closed because the function is [[continuous function|continuous]]. The graph of the [[Heaviside function]] from <nowiki>[-2,2]</nowiki> is not closed, because the function is not continuous.
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| image1    =cubicpoly.png
| alt1      = A cubic function

| image2    = Dirac distribution CDF.svg
| alt2      = The Heaviside function
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In [[mathematics]], the '''closed graph theorem''' is a basic result which characterizes [[continuous function]]s in terms of their [[graph of a function|graph]]s. There are several versions of the theorem.

== The closed graph theorem ==

In mathematics, there are several results known as the "closed graph theorem".

=== In point-set topology ===

For any function {{nowrap|''T'' : ''X'' &rarr; ''Y''}}, we define the ''graph'' of ''T'' to be the set

:<math>\lbrace (x,y) \in X\times Y \mid Tx=y\rbrace.</math>

In [[point-set topology]], the closed graph theorem states the following:
If ''X'' is a [[topological space]] and ''Y'' is a [[Compact space|compact]] [[Hausdorff space]], then the graph of ''T'' is closed if and only if ''T'' is [[continuous function (topology)|continuous]].<ref>{{harvtxt|Munkres|2000}}, p. 171</ref>

=== In functional analysis ===

In [[functional analysis]], the closed graph theorem states the following:
If ''X'' and ''Y'' are [[Banach space|Banach spaces]], and {{nowrap|''T'' : ''X'' &rarr; ''Y''}} is a [[linear operator]], then ''T'' is continuous if and only if its graph is [[closed set|closed]] in {{nowrap|''X''&thinsp;&times;&thinsp;''Y''}} (with the [[product topology]]).

In the latter case we say that ''T'' is a [[closed linear operator|closed operator]]. Note that the operator is required to be everywhere-defined, i.e., the [[domain (mathematics)|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 (functional analysis)|open mapping theorem]]. In fact, the closed graph theorem, the open mapping theorem and the [[bounded inverse theorem]] are all [[logically equivalent|equivalent]].  This equivalence also serves to demonstrate the necessity 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.

The closed graph theorem can be reformulated as follows. If {{nowrap|''T'' : ''X'' &rarr; ''Y''}} is a linear operator between Banach spaces, then the following are equivalent:
#  For every sequence {''x''<sub>''n''</sub>} in ''X'', if the sequence {''x''<sub>''n''</sub>} converges in ''X'' to some element ''x'', then the sequence {''T''(''x''<sub>''n''</sub>)} in ''Y'' also converges, and its limit is ''T''(''x''). 
# For every sequence {''x''<sub>''n''</sub>} in ''X'', if the sequence {''x''<sub>''n''</sub>} converges in ''X'' to some element ''x'' and the sequence {''T''(''x''<sub>''n''</sub>)} in ''Y'' converges to some element ''y'', then {{nowrap|''y'' {{=}} ''T''(''x'')}}.

=== For set-valued functions ===
The [[closed graph theorem]] for set-valued functions<ref name="aliprantis">{{cite book|url=|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|last=Aliprantis|first=Charlambos|author2=Kim C. Border|publisher=Springer|year=1999|edition=3rd|chapter=Chapter 17}}</ref> says that, for a compact [[Hausdorff space|Hausdorff]] range space ''Y'', a set-valued function ''φ'':&nbsp;''X''→2<sup>''Y''</sup> has a closed graph if and only if it is [[upper hemicontinuous]] and ''φ''(''x'') is a closed set for all ''x''.

== Generalization ==

The closed graph theorem can be generalized to more abstract [[topological vector spaces]] in the following ways:

:A linear operator from a [[barrelled space]] ''X'' to a [[Fréchet space]] ''Y'' is [[continuous linear operator|continuous]] if and only if its graph is closed in the space ''X''&times;''Y'' equipped with the product topology.

and there is a version that does not require ''Y'' to be locally convex:<ref>{{harvtxt|Trèves|1995}}, p. 173</ref>

:If the graph of a linear map between two [[F-space]]s is closed then the map is continuous.

and an even more general version of the closed graph theorem is<ref>Treves p. 169</ref>

: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 <math>X \times Y</math> and ''u'' is any continuous map of ''G'' onto ''X'', then ''u'' is an open mapping. Under this condition, if <math>f : X \to Y</math> is a linear map whose graph is closed then ''f'' is continuous.

=== 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.<ref>Treves p. 549</ref> 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-space]]s over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states:<ref>Treves p. 549</ref>

:Let ''X'' and ''Y'' be locally convex Hausdorff spaces and let <math>u : X \to Y</math> be linear. 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 <math>X \times Y</math>, then ''u'' is continuous.

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space ''X'' is called a <math>K_{\sigma \delta}</math> 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 <math>K_{\sigma \delta}</math> space (that is, if there is a <math>K_{\sigma \delta}</math> 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 Frechet space is K-analytic as is the weak dual of a Frechet space. The generalized theorem states:<ref>Treves pp. 557-558</ref>

:Let ''X'' and ''Y'' be locally convex Hausdorff spaces and let <math>u : X \to Y</math> be linear. 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 <math>X \times Y</math>, then ''u'' is continuous.

==See also==
* [[Discontinuous linear map]]
* [[Webbed space]]

==Notes==
{{reflist}}

== References ==
* {{citation | last = Munkres | first = James | authorlink = James Munkres | title=Topology | edition=2nd | publisher=[[Upper Saddle River: Prentice Hall]] | year=2000 | isbn=0-13-181629-2 | pages=163–172}}
* {{citation | last = Trèves | first = François | authorlink = François Trèves | title=Topological Vector Spaces, Distributions and Kernels | publisher=[[Academic Press, Inc.]] | year=1995 | isbn=0-486-45352-9 | pages=169–173, 549–557}}
* {{citation | last = Folland | first = Gerald B. | authorlink = Gerald Folland | title=Real Analysis: Modern Techniques and Their Applications | edition=1st | publisher=[[John Wiley & Sons]] | year = 1984 | isbn=978-0-471-80958-6 }}
* {{citation| first=Walter|last=Rudin|authorlink=Walter Rudin|title=Functional analysis|publisher=Tata MacGraw-Hill|year=1973}}.
* {{planetmath reference|id=6472|title=Proof of closed graph theorem }}

{{Functional Analysis}}

[[Category:Theorems in functional analysis]]