Closed graph theorem (functional analysis)

<h2 id="explanation">Explanation</h2>
Let 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 be a linear operator between Banach spaces (or more generally Fréchet spaces). Then the continuity of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 means that 
 
 
 
 T
 
 x
 
 i
 
 
 →
 T
 x
 
 
 {\displaystyle Tx_{i}\to Tx}
 
 for each convergent sequence 
 
 
 
 
 x
 
 i
 
 
 →
 x
 
 
 {\displaystyle x_{i}\to x}
 
. On the other hand, the closedness of the graph of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 means that for each convergent sequence 
 
 
 
 
 x
 
 i
 
 
 →
 x
 
 
 {\displaystyle x_{i}\to x}
 
 such that 
 
 
 
 T
 
 x
 
 i
 
 
 →
 y
 
 
 {\displaystyle Tx_{i}\to y}
 
, we have 
 
 
 
 y
 =
 T
 x
 
 
 {\displaystyle y=Tx}
 
. Hence, the closed graph theorem says that in order to check the continuity of 
 
 
 
 T
 
 
 {\displaystyle T}
 
, one can show 
 
 
 
 T
 
 x
 
 i
 
 
 →
 T
 x
 
 
 {\displaystyle Tx_{i}\to Tx}
 
 under the additional assumption that 
 
 
 
 T
 
 x
 
 i
 
 
 
 
 {\displaystyle Tx_{i}}
 
 is convergent.
In fact, for the graph of T to be closed, it is enough that if 
 
 
 
 
 x
 
 i
 
 
 →
 0
 ,
 
 T
 
 x
 
 i
 
 
 →
 y
 
 
 {\displaystyle x_{i}\to 0,\,Tx_{i}\to y}
 
, then 
 
 
 
 y
 =
 0
 
 
 {\displaystyle y=0}
 
. Indeed, assuming that condition holds, if 
 
 
 
 (
 
 x
 
 i
 
 
 ,
 T
 
 x
 
 i
 
 
 )
 →
 (
 x
 ,
 y
 )
 
 
 {\displaystyle (x_{i},Tx_{i})\to (x,y)}
 
, then 
 
 
 
 
 x
 
 i
 
 
 −
 x
 →
 0
 
 
 {\displaystyle x_{i}-x\to 0}
 
 and 
 
 
 
 T
 (
 
 x
 
 i
 
 
 −
 x
 )
 →
 y
 −
 T
 x
 
 
 {\displaystyle T(x_{i}-x)\to y-Tx}
 
. Thus, 
 
 
 
 y
 =
 T
 x
 
 
 {\displaystyle y=Tx}
 
; i.e., 
 
 
 
 (
 x
 ,
 y
 )
 
 
 {\displaystyle (x,y)}
 
 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> In practice, this works like this: T is some operator on some function space. One shows T is continuous with respect to the <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution topology</a>; 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 § Example for an explicit example.

<h2 id="statement">Statement</h2>

Theorem—<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
If 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 is a linear operator between <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> (or more generally <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>), then the following are equivalent:

<ol><li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 is continuous.</li>
<li>The graph of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is closed in the <a href="/facts/Product_topology/0hg02IaX">product topology</a> on 
 
 
 
 X
 ×
 Y
 .
 
 
 {\displaystyle X\times Y.}
 
</li></ol>

The usual proof of the closed graph theorem employs the <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">open mapping theorem</a>. It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see <a href="/facts/Closed_graph_theorem/CK2ra00W">closed graph theorem § Relation to the open mapping theorem</a> (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 <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">Open mapping theorem (functional analysis) § Statement and proof</a>, 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 
 
 
 
 
 Γ
 
 T
 
 
 
 
 {\displaystyle \Gamma _{T}}
 
 of T is closed. Then 
 
 
 
 
 Γ
 
 T
 
 
 ≃
 
 Γ
 
 
 T
 
 −
 1
 
 
 
 
 
 
 {\displaystyle \Gamma _{T}\simeq \Gamma _{T^{-1}}}
 
 under 
 
 
 
 (
 x
 ,
 y
 )
 ↦
 (
 y
 ,
 x
 )
 
 
 {\displaystyle (x,y)\mapsto (y,x)}
 
. Hence, by the closed graph theorem, 
 
 
 
 
 T
 
 −
 1
 
 
 
 
 {\displaystyle T^{-1}}
 
 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 <a href="/facts/Unbounded_operator/5u2njqHt">unbounded operator</a>) exists and thus serves as a counterexample.

<h2 id="example">Example</h2>
The <a href="/facts/Hausdorff%25E2%2580%2593Young_inequality/uFlluIBg">Hausdorff–Young inequality</a> says that the Fourier transformation 
 
 
 
 
 
 
 ⋅
 ^
 
 
 
 :
 
 L
 
 p
 
 
 (
 
 
 R
 
 
 n
 
 
 )
 →
 
 L
 
 
 p
 ′
 
 
 
 (
 
 
 R
 
 
 n
 
 
 )
 
 
 {\displaystyle {\widehat {\cdot }}:L^{p}(\mathbb {R} ^{n})\to L^{p'}(\mathbb {R} ^{n})}
 
 is a well-defined bounded operator with operator norm one when 
 
 
 
 1
 
 /
 
 p
 +
 1
 
 /
 
 
 p
 ′
 
 =
 1
 
 
 {\displaystyle 1/p+1/p'=1}
 
. This result is usually proved using the <a href="/facts/Riesz%25E2%2580%2593Thorin_interpolation_theorem/fH5LKWnW">Riesz–Thorin interpolation theorem</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Here is how the argument would go. Let T denote the Fourier transformation. First we show 
 
 
 
 T
 :
 
 L
 
 p
 
 
 →
 Z
 
 
 {\displaystyle T:L^{p}\to Z}
 
 is a continuous linear operator for Z = the space of tempered distributions on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
. Second, we note that T maps the space of <a href="/facts/Schwarz_function/DLnyoAte">Schwarz functions</a> 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 
 
 
 
 
 L
 
 p
 
 
 ×
 
 L
 
 
 p
 ′
 
 
 
 
 
 {\displaystyle L^{p}\times L^{p'}}
 
 and 
 
 
 
 T
 :
 
 L
 
 p
 
 
 →
 
 L
 
 
 p
 ′
 
 
 
 
 
 {\displaystyle T:L^{p}\to L^{p'}}
 
 is defined but with unknown bounds. Since 
 
 
 
 T
 :
 
 L
 
 p
 
 
 →
 Z
 
 
 {\displaystyle T:L^{p}\to Z}
 
 is continuous, the graph of 
 
 
 
 T
 :
 
 L
 
 p
 
 
 →
 
 L
 
 
 p
 ′
 
 
 
 
 
 {\displaystyle T:L^{p}\to L^{p'}}
 
 is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, 
 
 
 
 T
 :
 
 L
 
 p
 
 
 →
 
 L
 
 
 p
 ′
 
 
 
 
 
 {\displaystyle T:L^{p}\to L^{p'}}
 
 is a bounded operator.

<h2 id="generalization">Generalization</h2>
<h3>Complete metrizable codomain</h3>
The closed graph theorem can be generalized from Banach spaces to more abstract <a href="/facts/Topological_vector_spaces/XRex1ChN">topological vector spaces</a> in the following ways.

Theorem—A linear operator from a <a href="/facts/Barrelled_space/r2H58TN3">barrelled space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous</a> if and only if its graph is closed.

<h4>Between F-spaces</h4>
There are versions that does not require 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to be locally convex.

Theorem—A linear map between two <a href="/facts/F-space/NBCcYfcU">F-spaces</a> is continuous if and only if its graph is closed.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

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

Theorem—If 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 is a linear map between two <a href="/facts/F-space/NBCcYfcU">F-spaces</a>, then the following are equivalent:

<ol><li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 is continuous.</li>
<li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 has a closed graph.</li>
<li>If 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 →
 x
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 T
 
 (
 
 x
 
 ∙
 
 
 )
 
 :=
 
 
 (
 
 T
 
 (
 
 x
 
 i
 
 
 )
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle T\left(x_{\bullet }\right):=\left(T\left(x_{i}\right)\right)_{i=1}^{\infty }}
 
 converges in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to some 
 
 
 
 y
 ∈
 Y
 ,
 
 
 {\displaystyle y\in Y,}
 
 then 
 
 
 
 y
 =
 T
 (
 x
 )
 .
 
 
 {\displaystyle y=T(x).}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>If 
 
 
 
 
 x
 
 ∙
 
 
 =
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 →
 0
 
 
 {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 T
 
 (
 
 x
 
 ∙
 
 
 )
 
 
 
 {\displaystyle T\left(x_{\bullet }\right)}
 
 converges in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to some 
 
 
 
 y
 ∈
 Y
 ,
 
 
 {\displaystyle y\in Y,}
 
 then 
 
 
 
 y
 =
 0.
 
 
 {\displaystyle y=0.}
 
</li></ol>

<h3>Complete pseudometrizable codomain</h3>
Every <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable topological space</a> is <a href="/facts/Pseudometrizable/uy1o2lhk">pseudometrizable</a>. A <a href="/facts/Pseudometrizable/uy1o2lhk">pseudometrizable</a> space is metrizable if and only if it is <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>.

Closed Graph Theorem<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>—Also, a closed linear map from a locally convex <a href="/facts/Ultrabarrelled_space/u9sXT7rc">ultrabarrelled space</a> into a complete <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable TVS</a> is continuous.

Closed Graph Theorem—A closed and bounded linear map from a locally convex <a href="/facts/Infrabarreled_space/SaoLAH83">infrabarreled space</a> into a complete <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable</a> locally convex space is continuous.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h3>Codomain not complete or (pseudo) metrizable</h3>

Theorem<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>—Suppose that 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 is a linear map whose graph is closed. 
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an inductive limit of <a href="/facts/Baire_space/HToekdT8">Baire</a> TVSs and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Webbed_space/ChtDnm0X">webbed space</a> then 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is continuous.

Closed Graph Theorem<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>—A closed surjective linear map from a complete <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable TVS</a> onto a locally convex <a href="/facts/Ultrabarrelled_space/u9sXT7rc">ultrabarrelled space</a> is continuous.

An even more general version of the closed graph theorem is

Theorem<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>—Suppose that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is any closed subspace of 
 
 
 
 X
 ×
 Y
 
 
 {\displaystyle X\times Y}
 
 and 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is any continuous map of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 onto 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is an open mapping.
Under this condition, if 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 is a linear map whose graph is closed then 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is continuous.

<h2 id="borel-graph-theorem">Borel graph theorem</h2>
Main article: <a href="/facts/Borel_Graph_Theorem/Sx1gYYXD">Borel Graph Theorem</a>
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.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> 
Recall that a topological space is called a <a href="/facts/Polish_space/CkUFcGap">Polish space</a> if it is a separable complete metrizable space and that a <a href="/facts/Souslin_space/CkUFcGap">Souslin space</a> 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 <a href="/facts/Lp-space/gbad0Rv1">Lp-spaces</a> over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. 
The <a href="/facts/Borel_graph_theorem/Sx1gYYXD">Borel graph theorem</a> states:

Borel Graph Theorem—Let 
 
 
 
 u
 :
 X
 →
 Y
 
 
 {\displaystyle u:X\to Y}
 
 be linear map between two <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the inductive limit of an arbitrary family of Banach spaces, if 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a Souslin space, and if the graph of 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a Borel set in 
 
 
 
 X
 ×
 Y
 ,
 
 
 {\displaystyle X\times Y,}
 
 then 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is continuous.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces.
A topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called a 
 
 
 
 
 K
 
 σ
 δ
 
 
 
 
 {\displaystyle K_{\sigma \delta }}
 
 if it is the countable intersection of countable unions of compact sets.
A Hausdorff topological space 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is called K-analytic if it is the continuous image of a 
 
 
 
 
 K
 
 σ
 δ
 
 
 
 
 {\displaystyle K_{\sigma \delta }}
 
 space (that is, if there is a 
 
 
 
 
 K
 
 σ
 δ
 
 
 
 
 {\displaystyle K_{\sigma \delta }}
 
 space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and a continuous map of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 onto 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
).
Every compact set is K-analytic so that there are non-separable K-analytic spaces. Also, every Polish, Souslin, and <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a> <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> is K-analytic as is the weak dual of a Frechet space. 
The generalized Borel graph theorem states:

Generalized Borel Graph Theorem<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>—Let 
 
 
 
 u
 :
 X
 →
 Y
 
 
 {\displaystyle u:X\to Y}
 
 be a linear map between two locally convex Hausdorff spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the inductive limit of an arbitrary family of Banach spaces, if 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a K-analytic space, and if the graph of 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is closed in 
 
 
 
 X
 ×
 Y
 ,
 
 
 {\displaystyle X\times Y,}
 
 then 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is continuous.

<h2 id="related-results">Related results</h2>
If 
 
 
 
 F
 :
 X
 →
 Y
 
 
 {\displaystyle F:X\to Y}
 
 is closed linear operator from a Hausdorff <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 into a Hausdorff finite-dimensional TVS 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 then 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is continuous.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Almost_open_linear_map/ta3sk8AY">Almost open linear map</a> – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Barrelled_space/r2H58TN3">Barrelled space</a> – Type of topological vector space</li>
<li><a href="/facts/Closed_graph/canolygI">Closed graph</a> – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Closed_linear_operator/l01mnNoJ">Closed linear operator</a></li>
<li><a href="/facts/Densely_defined_operator/8uGARqJR">Densely defined operator</a> – Function that is defined almost everywhere (mathematics)</li>
<li><a href="/facts/Discontinuous_linear_map/W1BSWvau">Discontinuous linear map</a></li>
<li><a href="/facts/Kakutani_fixed-point_theorem/qMQNEjzU">Kakutani fixed-point theorem</a> – Fixed-point theorem for set-valued functions</li>
<li><a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">Open mapping theorem (functional analysis)</a> – Condition for a linear operator to be open</li>
<li><a href="/facts/Ursescu_theorem/4LsjFGo8">Ursescu theorem</a> – Generalization of closed graph, open mapping, and uniform boundedness theorem</li>
<li><a href="/facts/Webbed_space/ChtDnm0X">Webbed space</a> – Space where open mapping and closed graph theorems hold</li></ul>

Notes

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<li>Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. <a href="/facts/Lecture_Notes_in_Mathematics/5DmSFMhJ">Lecture Notes in Mathematics</a>. Vol. 692. Berlin, New York, Heidelberg: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09096-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/4493665">4493665</a>.</li>
<li>Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-519-02224-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/8210342">8210342</a>.</li>
<li><a href="/facts/Gottfried_K%25C3%25B6the/y7A72q5g">Köthe, Gottfried</a> (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-64988-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498">0248498</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840293704">840293704</a>.</li>
<li>Kriegl, Andreas; Michor, Peter W. (1997). <a href="https://www.mat.univie.ac.at/~michor/apbookh-ams.pdf">The Convenient Setting of Global Analysis</a> (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-0780-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/37141279">37141279</a>.</li>
<li><a href="/facts/James_Munkres/Ybfa6fs4">Munkres, James R.</a> (2000). Topology (2nd ed.). <a href="/facts/Upper_Saddle_River%2c_NJ/Ps08aEYT">Upper Saddle River, NJ</a>: <a href="/facts/Prentice_Hall%2c_Inc/bynWAuat">Prentice Hall, Inc</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-181629-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/42683260">42683260</a>.</li>
<li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li>Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-29882-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/589250">589250</a>.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li>
<li>Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8247-8643-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/24909067">24909067</a>.</li>
<li><a href="/facts/Terence_Tao/Do68JKn6">Tao, Terence</a>, <a href="https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/">245B, Notes 9: The Baire category theorem and its Banach space consequences</a></li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li>
<li>Vogt, Dietmar (2000), <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf">Lectures on Fréchet spaces</a> (PDF)</li>
<li><a href="/facts/Albert_Wilansky/I6QRVtGa">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49353-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/849801114">849801114</a>.</li>
<li><a href="https://planetmath.org/ProofOfClosedGraphTheorem">"Proof of closed graph theorem"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Theorem 4 of Tao. NB: The Hausdorffness there is put to ensure the graph of a continuous map is closed. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Vogt 2000, Theorem 1.8. - Vogt, Dietmar (2000), Lectures on Fréchet spaces (PDF) <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf" target="_blank">https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Tao, Example 3 - Tao, Terence, 245B, Notes 9: The Baire category theorem and its Banach space consequences <a href="https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/" target="_blank">https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schaefer & Wolff 1999, p. 78. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Trèves (2006), p. 173 - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Rudin 1991, pp. 50–52. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Narici & Beckenstein 2011, pp. 474–476. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Narici & Beckenstein 2011, pp. 474–476. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Narici & Beckenstein 2011, p. 479-483. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Narici & Beckenstein 2011, pp. 474–476. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Trèves 2006, p. 169. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Trèves 2006, p. 549. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Trèves 2006, p. 549. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Trèves 2006, pp. 557–558. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Narici & Beckenstein 2011, p. 476. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
</ol>

Closed graph theorem (functional analysis) open-in-new

Closed graph theorem (functional analysis)