View source for Closed graph theorem (functional analysis)

{{Short description|Theorems connecting continuity to closure of graphs}}
In mathematics, particularly in [[Functional analysis|functional analysis]] and [[Topology|topology]], the '''closed graph theorem''' is a result connecting the [[Continuous function|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|linear function]] between two Banach spaces is continuous [[If and only if|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 [[Partial function|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|linear map]] <math>f : X \to Y</math> between two vector spaces endowed with topologies making them into [[Topological vector space|topological vector space]]s (TVSs). We will henceforth assume that <math>X</math> and <math>Y</math> are topological vector spaces, such as [[Banach space]]s for example, and that [[Cartesian product]]s, such as <math>X \times Y,</math> are endowed with the [[Product topology|product topology]]. 
The {{em|graph}} of this function is the subset 
<math display=block>\operatorname{graph}{\!(f)} = \{(x, f(x)) : x \in \operatorname{dom} f\},</math>
of <math>\operatorname{dom} (f) \times Y = X \times Y,</math> where <math>\operatorname{dom} f = X</math> denotes the function's [[Domain of a function|domain]].
The map <math>f : X \to Y</math> is said to have a '''[[Closed graph property|{{visible anchor|closed graph}}]]''' ('''in <math>X \times Y</math>''') if its graph <math>\operatorname{graph} f</math> is a [[Closed set|closed subset]] of product space <math>X \times Y</math> (with the usual [[Product topology|product topology]]). 
Similarly, <math>f</math> is said to have a '''{{visible anchor|sequentially closed graph}}''' if <math>\operatorname{graph} f</math> is a sequentially closed subset of <math>X \times Y.</math> 

A '''{{visible anchor|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|general topology]].

'''Partial functions'''

It is common in functional analysis to consider [[Partial function|partial function]]s, which are functions defined on a [[Dense set|dense]] [[Subset|subset]] of some space <math>X.</math> 
A partial function <math>f</math> is declared with the notation <math>f : D \subseteq X \to Y,</math> which indicates that <math>f</math> has prototype <math>f : D \to Y</math> (that is, its [[Domain of a function|domain]] is <math>D</math> and its [[Codomain|codomain]] is <math>Y</math>) and that <math>\operatorname{dom} f = D</math> is a dense subset of <math>X.</math> 
Since the domain is denoted by <math>\operatorname{dom} f,</math> it is not always necessary to assign a symbol (such as <math>D</math>) to a partial function's domain, in which case the notation <math>f : X \rightarrowtail Y</math> or <math>f : X \rightharpoonup Y</math> may be used to indicate that <math>f</math> is a partial function with codomain <math>Y</math> whose domain <math>\operatorname{dom} f</math> is a dense subset of <math>X.</math>{{sfn|Dolecki|Mynard| 2016|pp=4-5}} 
A '''densely defined linear operator''' between vector spaces is a partial function <math>f : D \subseteq X \to Y</math> whose domain <math>D</math> is a dense vector subspace of a TVS <math>X</math> such that <math>f : D \to Y</math> is a [[Linear map|linear map]]. 
A prototypical example of a partial function is the derivative operator, which is only defined on the space <math>D := C^1([0, 1])</math> of once continuously differentiable functions, a [[Dense set|dense subset]] of the space <math>X := C([0, 1])</math> 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 <math>f</math> is (as before) the set 
<math display=inline>\operatorname{graph}{\!(f)} = \{(x, f(x)) : x \in \operatorname{dom} f\}.</math> 
However, one exception to this is the definition of "closed graph". A {{em|partial}} function <math>f : D \subseteq X \to Y</math> is said to have a '''closed graph''' (respectively, a '''sequentially closed graph''') if <math>\operatorname{graph} f</math> is a closed (respectively, sequentially closed) subset of <math>X \times Y</math> in the [[Product topology|product topology]]; importantly, note that the product space is <math>X \times Y</math> and {{em|not}} <math>D \times Y = \operatorname{dom} f \times Y</math> as it was defined above for ordinary functions.<ref group=note>In contrast, when <math>f : D \to Y</math> is considered as an ordinary function (rather than as the partial function <math>f : D \subseteq X \to Y</math>), then "having a closed graph" would instead mean that <math>\operatorname{graph} f</math> is a closed subset of <math>D \times Y.</math> If <math>\operatorname{graph} f</math> is a closed subset of <math>X \times Y</math> then it is also a closed subset of <math>\operatorname{dom} (f) \times Y</math> although the converse is not guaranteed in general.</ref>

===Closable maps and closures===

A linear operator <math>f : D \subseteq X \to Y</math> is '''{{visible anchor|closable}} in <math>X \times Y</math>''' if there exists a {{em|vector subspace}} <math>E \subseteq X</math>  containing <math>D</math> and a function (resp. multifunction) <math>F : E \to Y</math> whose graph is equal to the closure of the set <math>\operatorname{graph} f</math> in <math>X \times Y.</math> Such an <math>F</math> is called a '''closure of <math>f</math> in <math>X \times Y</math>''', is denoted by <math>\overline{f},</math> and necessarily extends <math>f.</math>

If <math>f : D \subseteq X \to Y</math> is a closable linear operator then a '''{{visible anchor|core}}''' or an '''{{visible anchor|essential domain}}''' of <math>f</math> is a subset <math>C \subseteq D</math> such that the closure in <math>X \times Y</math> of the graph of the restriction <math>f\big\vert_C : C \to Y</math> of <math>f</math> to <math>C</math> is equal to the closure of the graph of <math>f</math> in <math>X \times Y</math> (i.e. the closure of <math>\operatorname{graph} f</math> in <math>X \times Y</math> is equal to the closure of <math>\operatorname{graph} f\big\vert_C</math> in <math>X \times Y</math>).

===Characterizations of closed graphs (general topology)===

Throughout, let <math>X</math> and <math>Y</math> be topological spaces and <math>X \times Y</math> is endowed with the product topology.

====Function with a closed graph====

{{Main|Closed graph property}}

If <math>f : X \to Y</math> is a function then it is said to have a '''{{em|[[Closed graph property|closed graph]]}}''' if it satisfies any of the following are equivalent conditions:
<ol>
<li>(Definition): The graph <math>\operatorname{graph} f</math> of <math>f</math> is a closed subset of <math>X \times Y.</math></li>
<li>For every <math>x \in X</math> and [[Net (mathematics)|net]] <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> if <math>y \in Y</math> is such that the net <math>f\left(x_{\bull}\right) = \left(f\left(x_i\right)\right)_{i \in I} \to y</math> in <math>Y</math> then <math>y = f(x).</math>{{sfn|Narici|Beckenstein|2011|pp=459-483}}
* Compare this to the definition of continuity in terms of nets, which recall is the following: for every <math>x \in X</math> and net <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> <math>f\left(x_{\bull}\right) \to f(x)</math> in <math>Y.</math>
* Thus to show that the function <math>f</math> has a closed graph, it ''may'' be assumed that <math>f\left(x_{\bull}\right)</math> converges in <math>Y</math> to some <math>y \in Y</math> (and then show that <math>y = f(x)</math>) while to show that <math>f</math> is continuous, it may ''not'' be assumed that <math>f\left(x_{\bull}\right)</math> converges in <math>Y</math> to some <math>y \in Y</math> and instead, it must be proven that this is true (and moreover, it must more specifically be proven that <math>f\left(x_{\bull}\right)</math> converges to <math>f(x)</math> in <math>Y</math>).</li>
</ol>

and if <math>Y</math> is a [[Hausdorff space|Hausdorff]] [[Compact space|compact space]] then we may add to this list: 
<ol start=3>
<li><math>f</math> is continuous.{{sfn|Munkres|2000|p=171}}</li>
</ol>

and if both <math>X</math> and <math>Y</math> are [[First-countable space|first-countable]] spaces then we may add to this list: 
<ol start=4>
<li><math>f</math> has a sequentially closed graph in <math>X \times Y.</math></li>
</ol>

'''Function with a sequentially closed graph'''

If <math>f : X \to Y</math> is a function then the following are equivalent:
<ol>
<li><math>f</math> has a sequentially closed graph in <math>X \times Y.</math></li>
<li>Definition: the graph of <math>f</math> is a sequentially closed subset of <math>X \times Y.</math></li>
<li>For every <math>x \in X</math> and sequence <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> if <math>y \in Y</math> is such that the net <math>f\left(x_{\bull}\right) := \left(f\left(x_i\right)\right)_{i=1}^{\infty} \to y</math> in <math>Y</math> then <math>y = f(x).</math>{{sfn|Narici|Beckenstein|2011|pp=459-483}}</li>
</ol>

===Basic properties of maps with closed graphs===

Suppose <math>f : D(f) \subseteq X \to Y</math> is a linear operator between [[Banach space]]s.

<ul>
<li>If <math>A</math> is closed then <math>A - s \operatorname{Id}_{D(f)}</math> is closed where <math>s</math> is a scalar and <math>\operatorname{Id}_{D(f)}</math> is the [[Identity function|identity function]].</li>
<li>If <math>f</math> is closed, then its kernel (or nullspace) is a closed vector subspace of <math>X.</math></li>
<li>If <math>f</math> is closed and [[Injective function|injective]] then its [[Inverse function|inverse]] <math>f^{-1}</math> is also closed.</li>
<li>A linear operator <math>f</math> admits a closure if and only if for every <math>x \in X</math> and every pair of sequences <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> and <math>z_{\bull} = \left(z_i\right)_{i=1}^{\infty}</math> in <math>D(f)</math> both converging to <math>x</math> in <math>X,</math> such that both <math>f\left(x_{\bull}\right) = \left(f\left(x_i\right)\right)_{i=1}^{\infty}</math> and <math>f\left(z_{\bull}\right) = \left(f\left(z_i\right)\right)_{i=1}^{\infty}</math> converge in <math>Y,</math> one has <math>\lim_{i \to \infty} f\left(x_i\right) = \lim_{i \to \infty} f\left(z_i\right).</math></li>
</ul>

==Examples and counterexamples==

===Continuous but not closed maps===

<ul>
<li>Let <math>X</math> denote the real numbers <math>\R</math> with the usual [[Euclidean topology]] and let <math>Y</math> denote <math>\R</math> with the indiscrete topology (where <math>Y</math> is {{em|not}} Hausdorff and that every function valued in <math>Y</math> is continuous).  
Let <math>f : X \to Y</math> be defined by <math>f(0) = 1</math> and <math>f(x) = 0</math> for all <math>x \neq 0.</math> 
Then <math>f : X \to Y</math> is continuous but its graph is ''not'' closed in <math>X \times Y.</math>{{sfn|Narici|Beckenstein|2011|pp=459-483}}</li>
<li>If <math>X</math> is any space then the identity map <math>\operatorname{Id} : X \to X</math> is continuous but its graph, which is the diagonal <math>\operatorname{graph} \operatorname{Id} = \{ (x, x) : x \in X \},</math> is closed in <math>X \times X</math> if and only if <math>X</math> is Hausdorff.{{sfn|Rudin|1991|p=50}}  In particular, if <math>X</math> is not Hausdorff then <math>\operatorname{Id} : X \to X</math> is continuous but ''not'' closed.</li>
<li>If <math>f : X \to Y</math> is a continuous map whose graph is not closed then <math>Y</math> is ''not'' a Hausdorff space.</li>
</ul>

===Closed but not continuous maps===

<ul>
<li>
If <math>(X, \tau)</math> is a Hausdorff TVS and <math>\nu</math> is a vector topology on <math>X</math> that is strictly finer than <math>\tau,</math> then the identity map <math>\operatorname{Id} : (X, \tau) \to (X, \nu)</math> a closed discontinuous linear operator.{{sfn|Narici|Beckenstein|2011|p=480}}
</li>
<li>
Consider the [[Derivative|derivative]] operator <math>A = \frac{d}{d x}</math> where <math>X = Y = C([a, b]).</math>is the Banach space of all [[Continuous function|continuous function]]s on an [[Interval (mathematics)|interval]] <math>[a, b].</math>
If one takes its domain <math>D(f)</math> to be <math>C^1([a, b]),</math> then <math>f</math> is a closed operator, which is not bounded.<ref>{{Cite book|title=Introductory Functional Analysis With Applications|last=Kreyszig|first=Erwin|publisher=John Wiley & Sons. Inc.|year=1978|isbn=0-471-50731-8|location=USA|pages=294}}</ref> 
On the other hand, if <math>D(f)</math> is the space <math>C^\infty([a, b])</math> of smooth functions scalar valued functions then <math>f</math> will no longer be closed, but it will be closable, with the closure being its extension defined on <math>C^1([a, b]).</math>
</li>
<li>
Let <math>X</math> and <math>Y</math> both denote the real numbers <math>\R</math> with the usual [[Euclidean topology]]. Let <math>f : X \to Y</math> be defined by <math>f(0) = 0</math> and <math>f(x) = \frac{1}{x}</math> for all <math>x \neq 0.</math> Then <math>f : X \to Y</math> has a closed graph (and a sequentially closed graph) in <math>X \times Y = \R^2</math> but it is ''not'' continuous (since it has a discontinuity at <math>x = 0</math>).{{sfn|Narici|Beckenstein|2011|pp=459-483}}</li>
<li>Let <math>X</math> denote the real numbers <math>\R</math> with the usual [[Euclidean topology]], let <math>Y</math> denote <math>\R</math> with the discrete topology, and let <math>\operatorname{Id} : X \to Y</math> be the identity map (i.e. <math>\operatorname{Id}(x) := x</math> for every <math>x \in X</math>). Then <math>\operatorname{Id} : X \to Y</math> is a [[Linear map|linear map]] whose graph is closed in <math>X \times Y</math> but it is clearly ''not'' continuous (since singleton sets are open in <math>Y</math> but not in <math>X</math>).{{sfn|Narici|Beckenstein|2011|pp=459-483}}
</li>
</ul>

==Closed graph theorems==

===Between Banach spaces===
{{Math theorem|name=Closed Graph Theorem for Banach spaces|math_statement=
If <math>T : X \to Y</math> is an everywhere-defined linear operator between [[Banach space]]s, then the following are equivalent:
# <math>T</math> is continuous. 
# <math>T</math> is closed (that is, the graph of <math>T</math> is closed in the [[Product topology|product topology]] on <math>X \times Y).</math> 
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to x</math> in <math>X</math> then <math>T\left(x_{\bull}\right) := \left(T\left(x_i\right)\right)_{i=1}^{\infty} \to T(x)</math> in <math>Y.</math> 
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to 0</math> in <math>X</math> then <math>T\left(x_{\bull}\right) \to 0</math> in <math>Y.</math> 
# 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)</math> converges in <math>Y</math> to some <math>y \in Y,</math> then <math>y = T(x).</math>
# 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>
}}

The operator is required to be '''everywhere-defined''', that is, the [[Domain of a function|domain]] <math>D(T)</math> of <math>T</math> is <math>X.</math>  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 <math>C([0, 1]),</math> whose domain is a strict subset of <math>C([0, 1]).</math>

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|bounded inverse theorem]] are all equivalent. 
This equivalence also serves to demonstrate the importance of <math>X</math> and <math>Y</math> 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.

{{Math theorem|name=Theorem|math_statement=
A linear operator from a [[Barrelled space|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|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|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|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|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==


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-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:

{{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|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}}

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* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn|Wilansky|2013|p=}} -->
* {{planetmath reference|urlname=ProofOfClosedGraphTheorem|title=Proof of closed graph theorem }}

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

[[Category:Theorems in functional analysis]]

{{Sourceattribution|Closed graph theorem (functional analysis)}}