Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.^[1]^[2] A function $f : X \to Y$ between topological spaces has a closed graph if its graph is a closed subset of the product space $X \times Y$ . A related property is open graph.^[3]

This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

Definitions edit

Graphs and set-valued functions edit

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) }

.

Notation: If

Y

is a set then the power set of

Y

, which is the set of all subsets of

Y

, is denoted by

2 Y

or

𝒫(Y)

.

Definition: If

X

and

Y

are sets, a set-valued function in

Y

on

X

(also called a

Y

-valued multifunction on

X

) is a function

F : X \to 2 Y

with domain

X

that is valued in

2 Y

. That is,

F

is a function on

X

such that for every

x \in X

,

F (x)

is a subset of

Y

.

Some authors call a function $F : X \to 2 Y$ a set-valued function only if it satisfies the additional requirement that $F (x)$ is not empty for every $x \in X$ ; this article does not require this.

Definition and notation: If

F : X \to 2 Y

is a set-valued function in a set

Y

then the graph of

F

is the set

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

.

Definition: A function

f : X \to Y

can be canonically identified with the set-valued function

F : X \to 2 Y

defined by

F (x) := {f (x) }

for every

x \in X

, where

F

is called the canonical set-valued function induced by (or associated with)

f

.

Note that in this case, $Gr f = Gr F$ .

Open and closed graph edit

We give the more general definition of when a $Y$ -valued function or set-valued function defined on a subset $S$ of $X$ has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace $S$ of a topological vector space $X$ (and not necessarily defined on all of $X$ ). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

Assumptions: Throughout,

X

and

Y

are topological spaces,

S \subseteq X

, and

f

is a

Y

-valued function or set-valued function on

S

(i.e.

f : S \to Y

or

f : S \to 2 Y

).

X \times Y

will always be endowed with the product topology.

Definition:^[4] We say that

f

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

f

,

Gr f

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

X \times Y

when

X \times Y

is endowed with the product topology. If

S = X

or if

X

is clear from context then we may omit writing "in

X \times Y

"

Observation: If

g : S \to Y

is a function and

G

is the canonical set-valued function induced by

g

(i.e.

G : S \to 2 Y

is defined by

G (s) := {g (s) }

for every

s \in S

) then since

Gr g = Gr G

,

g

has a closed (resp. sequentially closed, open, sequentially open) graph in

X \times Y

if and only if the same is true of

G

.

Closable maps and closures edit

Definition: We say that the function (resp. set-valued function)

f

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

D \subseteq X

containing

S

and a function (resp. set-valued function)

F : D \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

.

Additional assumptions for linear maps: If in addition, $S$ , $X$ , and $Y$ are topological vector spaces and $f : S \to Y$ is a linear map then to call $f$ closable we also require that the set $D$ be a vector subspace of $X$ and the closure of $f$ be a linear map.

Definition: If

f

is closable on

S

then a core or essential domain of

f

is a subset

D \subseteq S

such that the closure in

X \times Y

of the graph of the restriction

f | D : D \to Y

of

f

to

D

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 | D

in

X \times Y

).

Closed maps and closed linear operators edit

Definition and notation: When we write

f : D (f) \subseteq X \to Y

then we mean that

f

is a

Y

-valued function with domain

D (f)

where

D (f) \subseteq X

. If we say that

f : D (f) \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 (f) \times Y

).

When reading literature in functional analysis, if $f : X \to Y$ is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then " $f$ is closed" will almost always means the following:

Definition: A map

f : X \to Y

is called closed if its graph is closed in

X \times Y

. In particular, the term "closed linear operator" will almost certainly refer to a linear map whose graph is closed.

Otherwise, especially in literature about point-set topology, " $f$ is closed" may 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 edit

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);^[4]
- 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;^[5]

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 $y = f (x)$ ;^[4]

set-valued function with a closed graph

If $F : X \to 2 Y$ is a set-valued function between topological spaces $X$ and $Y$ then the following are equivalent:

$F$ has a closed graph (in $X \times Y$ );
(definition) the graph of $F$ is a closed subset of $X \times Y$ ;

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

$F$ is upper hemicontinuous and $F (x)$ is a closed subset of $Y$ for all $x \in X$ ;^[6]

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

for all $x \in X$ , $y \in Y$ , and sequences $x • = (x i) \infty i =1$ in $X$ and $y • = (y i) \infty i =1$ in $Y$ such that $x • \to x$ in $X$ and $y • \to y$ in $Y$ , and $y i \in F (x i)$ for all $i$ , then $y \in F (x)$ .^{[citation needed]}

Sufficient conditions for a closed graph edit

If f : X → Y is a continuous function between topological spaces and if Y is Hausdorff then f has a closed graph in X × Y.^[4]
- Note that if $f : X \to Y$ is a function between Hausdorff topological spaces then it is possible for $f$ to have a closed graph in $X \times Y$ but not be continuous.

Closed graph theorems: When a closed graph implies continuity edit

Conditions that guarantee that a function with a closed graph is necessarily continuous are called closed graph theorems. Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous.

If $f : X \to Y$ is a function between topological spaces whose graph is closed in $X \times Y$ and if $Y$ is a compact space then $f : X \to Y$ is continuous.^[4]

Examples edit

Continuous but not closed maps edit

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 $X \times Y$ .^[4]
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.^[7] 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 edit

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 $f(x) = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/x$ 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 $x = 0$ ).^[4]
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 $X$ ).^[4]
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 $Id : (X, 𝜏) \to (X, 𝜐)$ is a closed discontinuous linear operator.^[8]

Closed linear operators edit

Every continuous linear operator valued in a Hausdorff topological vector space (TVS) has a closed graph and recall that a linear operator between two normed spaces is continuous if and only if it is bounded.

Definition: If

X

and

Y

are topological vector spaces (TVSs) then we call a linear map

f : D (f) \subseteq X \to Y

a closed linear operator if its graph is closed in

X \times Y

.

Closed graph theorem edit

The closed graph theorem states that any closed linear operator $f : X \to Y$ between two F-spaces (such as Banach spaces) is continuous, where recall that if $X$ and $Y$ are Banach spaces then $f : X \to Y$ being continuous is equivalent to $f$ being bounded.

Basic properties edit

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 $lim i \to \infty fx i = lim i \to \infty fy i$ .

Example edit

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.^[9] On the other hand if $D (f) = C \infty ([a, b])$ , then $f$ will no longer be closed, but it will be closable, with the closure being its extension defined on $C 1 ([a, b])$ .

References edit

^ Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.
^ Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.
^ Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.
^ ^a ^b ^c ^d ^e ^f ^g ^h Narici & Beckenstein 2011, pp. 459–483.
^ Munkres 2000, p. 171.
^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
^ Rudin p.50
^ Narici & Beckenstein 2011, p. 480.
^ Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8.

Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
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.
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.
Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

[1] Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.

[2] Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.

[:2-3] Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.

[FOOTNOTENariciBeckenstein2011459–483-4] ^ ^a ^b ^c ^d ^e ^f ^g ^h Narici & Beckenstein 2011, pp. 459–483.

[FOOTNOTEMunkres2000171-5] Munkres 2000, p. 171.

[aliprantis-6] Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.

[7] Rudin p.50

[FOOTNOTENariciBeckenstein2011480-8] Narici & Beckenstein 2011, p. 480.

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

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[2]

[3]

[4]

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[6]

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[9]

Closed graph property

Contents

Definitions edit

Graphs and set-valued functions edit

Open and closed graph edit

Closable maps and closures edit

Closed maps and closed linear operators edit

Characterizations edit

Sufficient conditions for a closed graph edit

Closed graph theorems: When a closed graph implies continuity edit

Examples edit

Continuous but not closed maps edit

Closed but not continuous maps edit

Closed linear operators edit

Closed graph theorem edit

Basic properties edit

Example edit

See also edit

References edit