Closed graph

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

Graphs and multifunctions

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 then a

Y

-valued multifunction on

X

, also called a set-valued function in

Y

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 multifunction 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 multifunction 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 multifunction

F : X \to 2 Y

defined by

F (x) := {f (x)

} for every

x \in X

, where

F

is called the canonical multifunction induced by (or associated with)

f

.

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

Open and closed graph

We give the more general definition of when a $Y$ -valued function or multifunction 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 multifunction 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 multifunction 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

Definition: We say that the function (resp. multifunction)

f

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

D \subseteq X

containing

S

and a function (resp. multifunction)

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

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

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]

Multifunction 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 $y \in F (x)$ .

Sufficient conditions for a closed graph

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

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

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 $X \times Y$ .[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

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 $f (x) = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} 1 / x$ 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 $Id : (X, 𝜏) \to (X, 𝜐)$ a closed discontinuous linear operator.[8]

Closed linear operators

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

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

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

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 {{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 $C 1 ([a, b])$ .

References

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. 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. ISSN 0304-4068.
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 (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.
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. 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. 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. 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. 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.

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[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. 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. ISSN 0304-4068.

[FOOTNOTENariciBeckenstein2011459-483-4] 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.

Closed graph

Definitions

Graphs and multifunctions

Open and closed graph

Closable maps and closures

Closed maps and closed linear operators

Characterizations

Sufficient conditions for a closed graph

Closed graph theorems: When a closed graph implies continuity

Examples

Continuous but not closed maps

Closed but not continuous maps

Closed linear operators

Closed graph theorem

Basic properties

Example

See also

References