View source for Closed graph theorem

{{Short description|Theorem relating continuity to graphs}}
{{About|closed graph theorems in [[general topology]]|the closed graph theorem in [[functional analysis]]|Closed graph theorem (functional analysis)}}
{{multiple image
| footer = The graph of the [[cubic function]] <math>f(x) = x^3 - 9x</math> on the interval <math>[-4, 4]</math> is closed because the function is [[Continuous function|continuous]]. The graph of the [[Heaviside function]] on <math>[-2, 2]</math> is not closed, because the function is not continuous.
| width     = 200
| image1    = cubicpoly.png
| alt1      = A cubic function
| image2    = Dirac distribution CDF.svg
| alt2      = The Heaviside function
}}
In [[mathematics]], the '''closed graph theorem''' may refer to one of several basic results characterizing [[continuous function]]s in terms of their [[graph of a function|graph]]s. 
Each gives conditions when functions with [[closed graph]]s are necessarily continuous.

A blog post<ref name="Tao">{{cite web | url=https://terrytao.wordpress.com/2012/11/20/the-closed-graph-theorem-in-various-categories/ | title=The closed graph theorem in various categories | date=21 November 2012 }}</ref> by [[Terence Tao|T. Tao]] lists several closed graph theorems throughout mathematics.

== Graphs and maps with closed graphs ==
{{Main|Closed graph}}

If <math>f : X \to Y</math> is a map between [[topological space]]s then the '''graph''' of <math>f</math> is the set <math>\Gamma_f := \{ (x, f(x)) : x \in X \}</math> or equivalently,
<math display=block>\Gamma_f := \{ (x, y) \in X \times Y : y = f(x) \}</math>
It is said that '''the graph of <math>f</math> is closed''' if <math>\Gamma_f</math> is a [[closed set|closed subset]] of <math>X \times Y</math> (with the [[product topology]]).

Any continuous function into a [[Hausdorff space]] has a closed graph (see {{section link||Closed_graph_theorem_in_point-set_topology}})

Any linear map, <math>L : X \to Y,</math> between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) <math>L</math> is sequentially continuous in the sense of the product topology, then the map <math>L</math> is continuous and its graph, {{math|Gr ''L''}}, is necessarily closed. Conversely, if <math>L</math> is such a linear map with, in place of (1a), the graph of <math>L</math> is (1b) known to be closed in the Cartesian product space <math>X \times Y</math>, then <math>L</math> is continuous and therefore necessarily sequentially continuous.{{sfn|Rudin|1991|p=51-52}} 

=== Examples of continuous maps that do ''not'' have a closed graph ===

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>\Gamma_{\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 does {{em|not}} have a closed graph. 

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 note that <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 {{em|not}} closed in <math>X \times Y</math>.{{sfn|Narici|Beckenstein|2011|pp=459-483}}

== Closed graph theorem in point-set topology ==

In [[point-set topology]], the closed graph theorem states the following:

{{Math theorem
| name = Closed graph theorem{{sfn|Munkres|2000|pp=163–172}}
| math_statement = If <math>f : X \to Y</math> is a map from a [[topological space]] <math>X</math> into a [[Hausdorff space]] <math>Y,</math> then the graph of <math>f</math> is closed if <math>f : X \to Y</math> is [[Continuous function (topology)|continuous]]. The converse is true when <math>Y</math> is [[Compact space|compact]]. (Note that compactness and Hausdorffness do not imply each other.)
}}

{{Math proof|title=Proof|drop=hidden|proof=

First part: just note that the graph of <math>f</math> is the same as the pre-image <math>(f \times \operatorname{id}_Y)^{-1}(D)</math> where <math>D = \{ (y, y) \mid y \in Y \}</math> is the diagonal in <math>Y^2</math>.

Second part: 

For any open <math>V\subset Y</math> , we check <math>f^{-1}(V)</math> is open. So take any <math>x\in f^{-1}(V)</math> , we construct some open neighborhood <math>U</math> of <math>x</math> , such that <math>f(U)\subset V</math> .

Since the graph of <math>f</math> is closed, for every point <math>(x, y')</math> on the "vertical line at x", with <math>y'\neq f(x)</math> , draw an open rectangle <math>U_{y'}\times V_{y'}</math> disjoint from the graph of <math>f</math> . These open rectangles, when projected to the y-axis, cover the y-axis except at <math>f(x)</math> , so add one more set <math>V</math>.

Naively attempting to take <math>U:= \bigcap_{y'\neq f(x)} U_{y'}</math> would construct a set containing <math>x</math>, but it is not guaranteed to be open, so we use compactness here.

Since <math>Y</math> is compact, we can take a finite open covering of <math>Y</math> as <math>\{V, V_{y'_1}, ..., V_{y'_n}\}</math>.

Now take <math>U:= \bigcap_{i=1}^n U_{y'_i}</math>. It is an open neighborhood of <math>x</math>, since it is merely a finite intersection. We claim this is the open neighborhood of <math>U</math> that we want.

Suppose not, then there is some unruly <math>x'\in U</math> such that <math>f(x') \not\in V</math> , then that would imply <math>f(x')\in V_{y'_i}</math> for some <math>i</math> by open covering, but then <math>(x', f(x'))\in U\times V_{y'_i} \subset U_{y'_i}\times V_{y'_i}</math> , a contradiction since it is supposed to be disjoint from the graph of <math>f</math> .
}}

If ''X'', ''Y'' are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see {{section link||Relation_to_the_open_mapping_theorem}}.<!-- maybe the full version too? -->

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact <math>Y</math> is the real line, which allows the discontinuous function with closed graph <math>f(x) = \begin{cases}
\frac 1 x \text{ if }x\neq 0,\\
0\text{ else}
\end{cases}</math>.

Also, [[closed linear operator]]s in functional analysis (linear operators with closed graphs) are typically not continuous.

=== For set-valued functions ===

{{Math theorem
| name = Closed graph theorem for set-valued functions<ref name="aliprantis">{{cite book|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>
| math_statement = For a [[Hausdorff space|Hausdorff]] [[Compact space|compact]] range space <math>Y</math>, a set-valued function <math>F : X \to 2^Y</math> has a closed graph if and only if it is [[upper hemicontinuous]] and {{math|''F''(''x'')}} is a closed set for all <math>x \in X</math>.
}}

== In functional analysis ==
{{Main|Closed graph theorem (functional analysis)}}

If <math>T : X \to Y</math> is a linear operator between [[topological vector space]]s (TVSs) then we say that <math>T</math> is a '''[[closed linear operator|closed operator]]''' if the graph of <math>T</math> is closed in <math>X \times Y</math> when <math>X \times Y</math> is endowed with the product topology.

The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. 
The original result has been generalized many times. 
A well known version of the closed graph theorems is the following.

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

The theorem is a consequence of the [[open mapping theorem (functional analysis)|open mapping theorem]]; see {{section link|| Relation to the open mapping theorem}} below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).

== Relation to the open mapping theorem ==
Often, the closed graph theorems are obtained as corollaries of the [[open mapping theorem]]s in the following way.<ref name="Tao" /><ref>{{cite arXiv | eprint=2403.03904 | last1=Noll | first1=Dominikus | title=Topological spaces satisfying a closed graph theorem | date=2024 | class=math.GN }}</ref> Let <math>f : X \to Y</math> be any map. Then it factors as
:<math>f: X \overset{i}\to \Gamma_f \overset{q}\to Y</math>.
Now, <math>i</math> is the inverse of the projection <math>p: \Gamma_f \to X</math>. So, if the open mapping theorem holds for <math>p</math>; i.e., <math>p</math> is an open mapping, then <math>i</math> is continuous and then <math>f</math> is continuous (as the composition of continuous maps).

For example, the above argument applies if <math>f</math> is a linear operator between Banach spaces with closed graph, or if <math>f</math> is a map with closed graph between compact Hausdorff spaces.

== See also ==

* {{annotated link|Almost open linear map}}
* {{annotated link|Barrelled space}}
* {{annotated link|Closed graph}}
* {{annotated link|Closed linear 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}}
* {{annotated link|Zariski's main theorem}}

== Notes ==

{{reflist|group=note}}
{{reflist|group=proof}}

== References ==

{{reflist}}

== Bibliography ==

* {{Bourbaki Topological Vector Spaces}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{citation|last=Folland|first = Gerald B.|author-link=Gerald Folland|title=Real Analysis: Modern Techniques and Their Applications|edition=1st|publisher=[[John Wiley & Sons]]|year=1984|isbn=978-0-471-80958-6}}
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1983|p=}} -->
* {{Munkres Topology|edition=2}} <!-- {{sfn|Munkres|2000|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|pp=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces}}
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn|Wilansky|2013|p=}} -->
* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=}} -->
* {{planetmath reference|urlname=ProofOfClosedGraphTheorem|title=Proof of closed graph theorem }}

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

[[Category:Theorems in functional analysis]]