Closed graph theorem (functional analysis)

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

Definitions

Graphs and closed graphs

Definition and notation: The graph of a function

f : X \to Y

is the set

Gr f := { (x, f (x)) : x \in X} = { (x, y) \in X \times Y : y = f (x)

}.

Assumption: If

X

and

Y

are topological spaces then

X \times Y

will always be endowed with the product topology.

Definition: If

X

and

Y

are topological spaces,

D \subseteq X

, and

f : D \to Y

is a function, then we say that

f

has a closed graph (resp. sequentially closed graph) in $X \times Y$ if the graph of

f

,

Gr f

, is a closed (resp. sequentially closed) subset of

X \times Y

. If

D = X

or if

X

is clear from context then we may omit writing "in

X \times Y

"

Linear operators

Definition and notation:[1] A partial map, denoted by

f : X ↣ Y

, if a map from a subset of

X

, denoted by

dom f

, into

Y

. If we write

f : D \subseteq X \to Y

then we mean that

f : X ↣ Y

is a partial map and

dom f = D

.

Definition and notation: If we say that

f : D \subseteq X \to Y

is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of

f

is closed (resp. sequentially closed) in

X \times Y

(rather than in

D \times Y

).

Definition: If we say that

f : D \subseteq X \to Y

is a linear or a linear operator then we mean that

X

and

Y

are vector spaces,

D \subseteq X

is a vector subspace of

X

,

f : D \to Y

is a linear map.

Closed linear operators

Assumption: We will henceforth assume that

X

and

Y

are topological vector spaces (TVSs).

Definition: A linear operator

f : D \subseteq X \to Y

is called closed or a closed linear operator if its graph is closed in

X \times Y

.

Closable maps and closures

Definition: We say that a linear operator

f : D \subseteq X \to Y

is closable in $X \times Y$ if there exists a vector subspace

E \subseteq X

containing

S

and a function (resp. multifunction)

F : E \to Y

whose graph is equal to the closure of the set

Gr f

in

X \times Y

. Such an

F

is called a closure of $f$ in $X \times Y$ , is denoted by

f

, and necessarily extends

f

.

Definition: If

f : D \subseteq X \to Y

is closable linear operator then a core or essential domain of

f

is a subset

C \subseteq D

such that the closure in

X \times Y

of the graph of the restriction

f | C : C \to Y

of

f

to

C

is equal to the closure of the graph of

f

in

X \times Y

(i.e. the closure of

Gr f

in

X \times Y

is equal to the closure of

Gr f | C

in

X \times Y

).

Closed maps vs. closed linear operators

When reading literature in functional analysis, if $f : X \to Y$ is a linear map between topological vector spaces (TVSs) then " $f$ is closed" will almost always mean that its graph is closed. However, " $f$ is closed" may, especially in literature about point-set topology, instead mean the following:

Definition: A map

f : X \to Y

between topological spaces is called a closed map if the image of a closed subset of

X

is a closed subset of

Y

.

These two definitions of "closed map" are not equivalent. If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.

Characterizations of closed graphs (general topology)

Throughout, let $X$ and $Y$ be topological spaces.

Function with a closed graph

If $f : X \to Y$ is a function then the following are equivalent:

$f$ has a closed graph (in $X \times Y$ );
(definition) the graph of $f$ , $Gr f$ , is a closed subset of $X \times Y$ ;
for every x ∈ X and net x_• = (x_i)_{i ∈ I} in X such that x_• → x in X, if y ∈ Y is such that the net f(x_•) := (f(x_i))_{i ∈ I} → y in Y then y = f(x);[2]
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x \in X$ and net $x • = (x i) i \in I$ in $X$ such that $x • \to x$ in $X$ , $f (x •) \to f (x)$ in $Y$ .
- Thus to show that the function $f$ has a closed graph we may assume that $f (x •)$ converges in $Y$ to some $y \in Y$ (and then show that $y = f (x)$ ) while to show that $f$ is continuous we may not assume that $f (x •)$ converges in $Y$ to some $y \in Y$ and we must instead prove that this is true (and moreover, we must more specifically prove that $f (x •)$ converges to $f (x)$ in $Y$ ).

and if $Y$ is a Hausdorff compact space then we may add to this list:

$f$ is continuous;[3]

and if both $X$ and $Y$ are first-countable spaces then we may add to this list:

$f$ has a sequentially closed graph (in $X \times Y$ );

Function with a sequentially closed graph

If $f : X \to Y$ is a function then the following are equivalent:

$f$ has a sequentially closed graph (in $X \times Y$ );
(definition) the graph of $f$ is a sequentially closed subset of $X \times Y$ ;
for every $x \in X$ and sequence $x • = (x i) \infty i =1$ in $X$ such that $x • \to x$ in $X$ , if $y \in Y$ is such that the net $f (x •) := (f (x i)) \infty i =1 \to y$ in $Y$ then $y = f (x)$ ;[2]

Basic properties of maps with closed graphs

The following properties are easily checked for a linear operator $f : D (f) \subseteq X \to Y$ between Banach spaces:

If $A$ is closed then $A - λ Id D (f)$ is closed where $λ$ is a scalar and $Id D (f)$ is the identity function;
If $f$ is closed, then its kernel (or nullspace) is a closed vector subspace of $X$ ;
If $f$ is closed and injective then its inverse $f -1$ is also closed;
A linear operator $f$ admits a closure if and only if for every $x \in X$ and every pair of sequences $x • = (x i) \infty i =1$ and $y • = (y i) \infty i =1$ in $D (f)$ both converging to $x$ in $X$ , such that both $f (x •) = (f (x i)) \infty i =1$ and $f (y •) = (f (y i)) \infty i =1$ converge in $Y$ , one has lim $i \to \infty$ $fx i =$ lim $i \to \infty$ $fy i$ .

Examples and counterexamples

Continuous but not closed maps

Let $X$ denote the real numbers $ℝ$ with the usual Euclidean topology and let $Y$ denote $ℝ$ with the indiscrete topology (where note that $Y$ is not Hausdorff and that every function valued in $Y$ is continuous). Let $f : X \to Y$ be defined by $f (0) = 1$ and $f (x) = 0$ for all $x \neq 0$ . Then $f : X \to Y$ is continuous but its graph is not closed in $X \times Y$ .[2]
If $X$ is any space then the identity map $Id : X \to X$ is continuous but its graph, which is the diagonal $Gr Id := { (x, x) : x \in X$ }, is closed in $X \times X$ if and only if $X$ is Hausdorff.[4] In particular, if $X$ is not Hausdorff then $Id : X \to X$ is continuous but not closed.
If $f : X \to Y$ is a continuous map whose graph is not closed then $Y$ is not a Hausdorff space.

Closed but not continuous maps

Let $(X, 𝜏)$ be a Hausdorff TVS and let $𝜐$ be a vector topology on $X$ that is strictly finer than $𝜏$ . Then the identity map $Id : (X, 𝜏) \to (X, 𝜐)$ a closed discontinuous linear operator.[5]
Consider the derivative operator $A = d / dx$ where $X = Y = C ([a, b])$ is the Banach space of all continuous functions on an interval $[a, b]$ . If one takes its domain $D (f)$ to be $C 1 ([a, b])$ , then $f$ is a closed operator, which is not bounded.[6] On the other hand if {{math|1=D(f) = [[smooth function|C^∞([a, b])]]}}, then $f$ will no longer be closed, but it will be closable, with the closure being its extension defined on $C 1 ([a, b])$ .
Let $X$ and $Y$ both denote the real numbers $ℝ$ with the usual Euclidean topology. Let $f : X \to Y$ be defined by $f (0) = 0$ and $f (x) = 1 / x$ for all $x \neq 0$ . Then $f : X \to Y$ has a closed graph (and a sequentially closed graph) in $X \times Y = ℝ 2$ but it is not continuous (since it has a discontinuity at $x = 0$ ).[2]
Let $X$ denote the real numbers $ℝ$ with the usual Euclidean topology, let $Y$ denote $ℝ$ with the discrete topology, and let $Id : X \to Y$ be the identity map (i.e. $Id(x) := x$ for every $x \in X$ ). Then $Id : X \to Y$ is a linear map whose graph is closed in $X \times Y$ but it is clearly not continuous (since singleton sets are open in $Y$ but not in $X$ ).[2]

Closed graph theorems

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

Theorem — A linear operator from a barrelled space $X$ to a Fréchet space $Y$ is continuous if and only if its graph is closed.

and there are versions that does not require $Y$ to be locally convex:

Theorem[7][8] — A linear map between two F-spaces is continuous if and only if its graph is closed.

We restate this theorem and extend it with some conditions that can be used to determine if a graph is closed:

Theorem — If $T : X \to Y$ is a linear map between two F-spaces, then the following are equivalent:

$T$ is continuous;
$T$ has a closed graph;
If $(x i) \infty i =1 \to x$ in $X$ and if $(T (x i)) \infty i =1$ converges in $Y$ to some $y \in Y$ , then $y = T (x)$ ;[9]
If $(x i) \infty i =1 \to 0$ in $X$ and if $(T (x i)) \infty i =1$ converges in $Y$ to some $y \in Y$ , then $y = 0$

Theorem[10] — Suppose that $T : X \to Y$ is a linear map whose graph is closed. If $X$ is an inductive limit of Baire TVS and $Y$ is a webbed space then $T$ is continuous.

Closed Graph Theorem[11] — A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.

Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.

Closed Graph Theorem[11] — A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous.

An even more general version of the closed graph theorem is

Theorem[12] — 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 $X \times Y$ and $u$ is any continuous map of $G$ onto $X$ , then $u$ is an open mapping. Under this condition, if $T : X \to Y$ is a linear map whose graph is closed then $T$ is continuous.

Between Banach spaces

In functional analysis, the closed graph theorem states the following: If $X$ and $Y$ are Banach spaces, and $T : X \to Y$ is a linear operator, then $T$ is continuous if and only if its graph is closed in $X \times Y$ (with the product topology).

The closed graph theorem can be reformulated may be rewritten into a form that is more easily usable:

Closed Graph Theorem for Banach spaces — If T : X → Y is a linear operator between Banach spaces, then the following are equivalent:

$T$ is continuous.
$T$ is a closed operator (i.e. the graph of $T$ is closed).
If $(x i) \infty i =1 \to x$ in $X$ then $(T (x i)) \infty i =1 \to T (x)$ in $Y$ .
If $(x i) \infty i =1 \to 0$ in $X$ then $(T (x i)) \infty i =1 \to 0$ in $Y$ .
If $(x i) \infty i =1 \to x$ in $X$ and if $(T (x i)) \infty i =1$ converges in $Y$ to some $y \in Y$ , then $y = T (x)$ .
If $\left(x_{i}\right)_{i=1}^{\infty }\to 0$ in $X$ and if $(T (x i)) \infty i =1$ converges in $Y$ to some $y \in Y$ , then $y = 0$ .

Note that the operator is required to be everywhere-defined, that is, the 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. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the importance 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.

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.[13] 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:[13]

Borel Graph Theorem — Let $u : X \to Y$ be linear map between two locally convex Hausdorff spaces $X$ and $Y$ . 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 $X \times Y$ , then $u$ is continuous.

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

Definition: A topological space

X

is called a

K σ δ

if it is the countable intersection of countable unions of compact sets.

Definition: A Hausdorff topological space

Y

is called K-analytic if it is the continuous image of a

K σ δ

space (that is, if there is a

K σ δ

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 Borel graph theorem states:

Generalized Borel Graph Theorem[14] — Let $u : X \to Y$ be a linear map between two locally convex Hausdorff spaces $X$ and $Y$ . 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 $X \times Y$ , then $u$ is continuous.

Related results

Theorem[15] — If $F : X \to Y$ is closed linear operator from a Hausdorff locally convex TVS $X$ into a Hausdorff finite-dimensional TVS $Y$ then $F$ is continuous.

Notes

Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.CS1 maint: ref=harv (link)
Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.CS1 maint: ref=harv (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Dolecki, Szymon; Mynard, Frederic (May 13, 2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.CS1 maint: date and year (link)
Dubinsky, Ed (June 1, 1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. 720. Berlin New York: Springer-Verlag. ISBN 978-3-540-09504-0. OCLC 5126156.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Husain, Taqdir; Khaleelulla, S. M. (December 5, 1978). Written at Berlin Heidelberg. Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin New York: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
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Munkres, James R. (January 7, 2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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"Proof of closed graph theorem". PlanetMath.

References

Dolecki 2016, pp. 4-5.
Narici 2011, pp. 459-483.
Munkres 2000, p. 171.
Rudin 1991, p. 50.
Narici 2011, p. 480.
Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8.
Schaefer 1999, p. 78.
Trèves (1995), p. 173
Rudin 1991, pp. 50-52.
Narici 2011, p. 479-483.
Narici 2011, pp. 474-476.
Trèves 2006, p. 169.
Trèves 2006, p. 549.
Trèves 2006, pp. 557-558.
Narici 2011, p. 476.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[FOOTNOTEDolecki20164-5-1] Dolecki 2016, pp. 4-5.

[FOOTNOTENarici2011459-483-2] Narici 2011, pp. 459-483.

[FOOTNOTEMunkres2000171-3] Munkres 2000, p. 171.

[FOOTNOTERudin199150-4] Rudin 1991, p. 50.

[FOOTNOTENarici2011480-5] Narici 2011, p. 480.

[6] Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8.

[FOOTNOTESchaefer199978-7] Schaefer 1999, p. 78.

[8] Trèves (1995), p. 173

[FOOTNOTERudin199150-52-9] Rudin 1991, pp. 50-52.

[FOOTNOTENarici2011479-483-10] Narici 2011, p. 479-483.

[FOOTNOTENarici2011474-476-11] Narici 2011, pp. 474-476.

[FOOTNOTETrèves2006169-12] Trèves 2006, p. 169.

[FOOTNOTETrèves2006549-13] Trèves 2006, p. 549.

[FOOTNOTETrèves2006557-558-14] Trèves 2006, pp. 557-558.

[FOOTNOTENarici2011476-15] Narici 2011, p. 476.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-Valued Hahn-Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property