View source for Closed graph theorem (functional analysis)

{{Short description|Theorems connecting continuity to closure of graphs}}
{{About|closed graph theorems in [[functional analysis]]|other results with the same name|Closed graph theorem}}
In mathematics, particularly in [[functional analysis]], the '''closed graph theorem''' is a result connecting the [[continuous function|continuity]] of a [[linear operator]] to a topological property of their [[graph (function)|graph]].  Precisely, the theorem states that a linear operator between two [[Banach spaces]] is continuous [[if and only if]] the graph of the operator is closed (such an operator is called a [[closed linear operator]]; see also [[closed graph property]]).

An important question in functional analysis is whether a given linear operator is continuous (or bounded). The closed graph theorem gives one answer to that question.<!-- the following is a bit strange since the closed graph theorem doesn’t really apply to partially defined operators.

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

== Explanation ==
Let <math>T : X \to Y</math> be a linear operator between Banach spaces (or more generally Fréchet spaces). Then the continuity of <math>T</math> means that <math>Tx_i \to Tx</math> for each convergent sequence <math>x_i \to x</math>. On the other hand, the closedness of the graph of <math>T</math> means that for each convergent sequence <math>x_i \to x</math> such that <math>Tx_i \to y</math>, we have <math>y = Tx</math>. Hence, the closed graph theorem says that in order to check the continuity of <math>T</math>, one can show <math>T x_i \to Tx</math> under the additional assumption that <math>Tx_i</math> is convergent.

In fact, for the graph of ''T'' to be closed, it is enough that if <math>x_i \to 0, \, Tx_i \to y</math>, then <math>y = 0</math>. Indeed, assuming that condition holds, if <math>(x_i, Tx_i) \to (x, y)</math>, then <math>x_i - x \to 0</math> and <math>T(x_i - x) \to y - Tx</math>. Thus, <math>y = Tx</math>; i.e., <math>(x, y)</math> is in the graph of ''T''.

Note, to check the closedness of a graph, it’s not even necessary to use the norm topology: if the graph of ''T'' is closed in some topology coarser than the norm topology, then it is closed in the norm topology.<ref>Theorem 4 of Tao. NB: The Hausdorffness there is put to ensure the graph of a continuous map is closed.</ref> In practice, this works like this: ''T'' is some operator on some function space. One shows ''T'' is continuous with respect to the [[distribution (mathematics)|distribution topology]]; thus, the graph is closed in that topology, which implies closedness in the norm topology and then ''T'' is a bounded by the closed graph theorem (when the theorem applies). See {{section link||Example}} for an explicit example.

==Statement ==

{{Math theorem|math_statement= <ref>{{harvnb|Vogt|2000|loc=Theorem 1.8.}}</ref>
If <math>T : X \to Y</math> is a linear operator between [[Banach space]]s (or more generally [[Fréchet space]]s), then the following are equivalent:
# <math>T</math> is continuous. 
# The graph of <math>T</math> is closed in the [[product topology]] on <math>X \times Y.</math> 
}}

The usual proof of the closed graph theorem employs the [[Open mapping theorem (functional analysis)|open mapping theorem]]. It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see {{section link|closed_graph_theorem|Relation_to_the_open_mapping_theorem}} (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)

In fact, the open mapping theorem can in turn be deduced from the closed graph theorem as follows. As noted in {{section link|Open_mapping_theorem_(functional_analysis)|Statement_and_proof}}, it is enough to prove the open mapping theorem for a continuous linear operator that is bijective (not just surjective). Let ''T'' be such an operator. Then by continuity, the graph <math>\Gamma_T</math> of ''T'' is closed. Then <math>\Gamma_T \simeq \Gamma_{T^{-1}}</math> under <math>(x, y) \mapsto (y, x)</math>. Hence, by the closed graph theorem, <math>T^{-1}</math> is continuous; i.e., ''T'' is an open mapping.

Since the closed graph theorem is equivalent to the open mapping theorem, one knows that the theorem fails without the completeness assumption. But more concretely, an operator with closed graph that is not bounded (see [[unbounded operator]]) exists and thus serves as a counterexample.

== Example ==
The [[Hausdorff–Young inequality]] says that the Fourier transformation <math>\widehat{\cdot} : L^p(\mathbb{R}^n) \to L^{p'}(\mathbb{R}^n)</math> is a well-defined bounded operator with operator norm one when <math>1/p + 1/p' = 1</math>. This result is usually proved using the [[Riesz–Thorin interpolation theorem]] and is highly nontrivial. The closed graph theorem can be used to prove a soft version of this result; i.e., the Fourier transformation is a bounded operator with the unknown operator norm.<ref>{{harvnb|Tao|loc=Example 3}}</ref>

Here is how the argument would go. Let ''T'' denote the Fourier transformation. First we show <math>T : L^p \to Z</math> is a continuous linear operator for ''Z'' = the space of tempered distributions on <math> \mathbb{R}^n</math>. Second, we note that ''T'' maps the space of [[Schwarz function]]s to itself (in short, because smoothness and rapid decay transform to rapid decay and smoothness, respectively). This implies that the graph of ''T'' is contained in <math>L^p \times L^{p'}</math> and <math>T : L^p \to L^{p'}</math> is defined but with unknown bounds.{{clarify|where was the assumption <math>1/p + 1/p' = 1</math> is used?|date=July 2024}} Since <math>T : L^p \to Z</math> is continuous, the graph of <math>T : L^p \to L^{p'}</math> is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, <math>T : L^p \to L^{p'}</math> is a bounded operator.

== Generalization ==

===Complete metrizable codomain===

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

{{Math theorem|name=Theorem|math_statement=
A linear operator from a [[barrelled space]] <math>X</math> to a [[Fréchet space]] <math>Y</math> is [[Continuous linear operator|continuous]] if and only if its graph is closed. 
}}

====Between F-spaces====

There are versions that does not require <math>Y</math> to be locally convex.

{{Math theorem|name=Theorem|math_statement=
A linear map between two [[F-space]]s is continuous if and only if its graph is closed.{{sfn|Schaefer|Wolff|1999|p=78}}<ref>{{harvtxt|Trèves|2006}}, p. 173</ref>
}}

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

{{Math theorem|name=Theorem|math_statement=
If <math>T : X \to Y</math> is a linear map between two [[F-space]]s, then the following are equivalent:
# <math>T</math> is continuous.
# <math>T</math> has a closed graph.
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to x</math> in <math>X</math> and if <math>T\left(x_{\bull}\right) := \left(T\left(x_i\right)\right)_{i=1}^{\infty}</math> converges in <math>Y</math> to some <math>y \in Y,</math> then <math>y = T(x).</math>{{sfn|Rudin|1991|pp=50-52}} 
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to 0</math> in <math>X</math> and if <math>T\left(x_{\bull}\right)</math> converges in <math>Y</math> to some <math>y \in Y,</math> then <math>y = 0.</math>
}}

===Complete pseudometrizable codomain===

Every [[Metrizable space|metrizable topological space]] is [[pseudometrizable]]. A [[pseudometrizable]] space is metrizable if and only if it is [[Hausdorff space|Hausdorff]].

{{Math theorem|name=Closed Graph Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=
Also, a closed linear map from a locally convex [[ultrabarrelled space]] into a complete [[Metrizable topological vector space|pseudometrizable TVS]] is continuous. 
}}

{{Math theorem|name=Closed Graph Theorem|math_statement=
A closed and bounded linear map from a locally convex [[infrabarreled space]] into a complete [[Metrizable topological vector space|pseudometrizable]] locally convex space is continuous.{{sfn|Narici|Beckenstein|2011|pp=474-476}} 
}}

===Codomain not complete or (pseudo) metrizable===

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=479-483}}|math_statement=
Suppose that <math>T : X \to Y</math> is a linear map whose graph is closed. 
If <math>X</math> is an inductive limit of [[Baire space|Baire]] TVSs and <math>Y</math> is a [[webbed space]] then <math>T</math> is continuous.
}}

{{Math theorem|name=Closed Graph Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=
A closed surjective linear map from a complete [[Metrizable topological vector space|pseudometrizable TVS]] onto a locally convex [[ultrabarrelled space]] is continuous.
}}

An even more general version of the closed graph theorem is

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=169}}|math_statement=
Suppose that <math>X</math> and <math>Y</math> are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

:If <math>G</math> is any closed subspace of <math>X \times Y</math> and <math>u</math> is any continuous map of <math>G</math> onto <math>X,</math> then <math>u</math> is an open mapping.

Under this condition, if <math>T : X \to Y</math> is a linear map whose graph is closed then <math>T</math> is continuous.
}}

==Borel graph theorem==

{{Main|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.{{sfn|Trèves|2006|p=549}} 
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:

{{Math theorem|name=Borel Graph Theorem|math_statement=
Let <math>u : X \to Y</math> be linear map between two [[locally convex]] [[Hausdorff space|Hausdorff]] spaces <math>X</math> and <math>Y.</math> If <math>X</math> is the inductive limit of an arbitrary family of Banach spaces, if <math>Y</math> is a Souslin space, and if the graph of <math>u</math> is a Borel set in <math>X \times Y,</math> then <math>u</math> is continuous.{{sfn|Trèves|2006|p=549}}   
}}

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces.

A topological space <math>X</math> is called a <math>K_{\sigma\delta}</math> if it is the countable intersection of countable unions of compact sets.

A Hausdorff topological space <math>Y</math> 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 <math>X</math> and a continuous map of <math>X</math> onto <math>Y</math>).

Every compact set is K-analytic so that there are non-separable K-analytic spaces.  Also, every Polish, Souslin, and [[Reflexive space|reflexive]] [[Fréchet space]] is K-analytic as is the weak dual of a Frechet space. 
The generalized Borel graph theorem states:

{{Math theorem|name=Generalized Borel Graph Theorem{{sfn|Trèves|2006|pp=557-558}}|math_statement=
Let <math>u : X \to Y</math> be a linear map between two locally convex Hausdorff spaces <math>X</math> and <math>Y.</math> If <math>X</math> is the inductive limit of an arbitrary family of Banach spaces, if <math>Y</math> is a K-analytic space, and if the graph of <math>u</math> is closed in <math>X \times Y,</math> then <math>u</math> is continuous.
}}

==Related results==

If <math>F : X \to Y</math> is closed linear operator from a Hausdorff [[locally convex]] TVS <math>X</math> into a Hausdorff finite-dimensional TVS <math>Y</math> then <math>F</math> is continuous.{{sfn|Narici|Beckenstein|2011|p=476}}

==See also==

* {{annotated link|Almost open linear map}}
* {{annotated link|Barrelled space}}
* {{annotated link|Closed graph}}
* {{annotated link|Closed linear operator}}
* {{annotated link|Densely defined operator}}
* {{annotated link|Discontinuous linear map}}
* {{annotated link|Kakutani fixed-point theorem}}
* {{annotated link|Open mapping theorem (functional analysis)}}
* {{annotated link|Ursescu theorem}}
* {{annotated link|Webbed space}}

==References==

'''Notes'''

{{reflist|group=note}}

{{reflist}}

==Bibliography==

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* {{planetmath reference|urlname=ProofOfClosedGraphTheorem|title=Proof of closed graph theorem }}

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[[Category:Theorems in functional analysis]]