mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, closed graph is a property of functions. A function between

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s has a closed graph if its

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

is a

closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

of the

product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...

. A related property is open graph. 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

Definitions

Graphs and set-valued functions

:Definition and notation: The graph of a function is the set ::. :Notation: If is a set then the power set of , which is the set of all subsets of , is denoted by or . :Definition: If and are sets, a

set-valued function A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathe ...

in on (also called a -valued multifunction on ) is a function with domain that is valued in . That is, is a function on such that for every , is a subset of . :* Some authors call a function a set-valued function only if it satisfies the additional requirement that is not empty for every ; this article does not require this. :Definition and notation: If is a set-valued function in a set then the graph of is the set ::. :Definition: A function can be canonically identified with the set-valued function defined by for every , where is called the canonical set-valued function induced by (or associated with) . :*Note that in this case, .

Open and closed graph

We give the more general definition of when a -valued function or set-valued function defined on a ''subset'' of has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace of a

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(and not necessarily defined on all of ). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis. :Assumptions: Throughout, and are topological spaces, , and is a -valued function or set-valued function on (i.e. or ). will always be endowed with the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...

. :Definition: We say that has a closed graph in if the graph of , , is a closed subset of when is endowed with the product topology. If or if is clear from context then we may omit writing "in " Note that we may define an

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

graph, a sequentially closed graph, and a sequentially open graph in similar ways. :Observation: If is a function and is the canonical set-valued function induced by (i.e. is defined by for every ) then since , has a closed (resp. sequentially closed, open, sequentially open) graph in if and only if the same is true of .

Closable maps and closures

:Definition: We say that the function (resp. set-valued function) is closable in if there exists a subset containing and a function (resp. set-valued function) whose graph is equal to the closure of the set in . Such an is called a closure of in , is denoted by , and necessarily extends . :*Additional assumptions for linear maps: If in addition, , , and are topological vector spaces and is a

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...

then to call closable we also require that the set be a vector subspace of and the closure of be a linear map. :Definition: If is closable on then a core or essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).

Closed maps and closed linear operators

:Definition and notation: When we write then we mean that is a -valued function with domain where . If we say that is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of is closed (resp. sequentially closed) in (rather than in ). When reading literature in

, if is a linear map between topological vector spaces (TVSs) (e.g.

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

s) then " is closed" will almost always means the following: :Definition: A map is called closed if its graph is closed in . 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 In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

, " is closed" may instead mean the following: :Definition: A map between topological spaces is called a

closed map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...

if the image of a closed subset of is a closed subset of . 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 and be topological spaces. ;Function with a closed graph If is a function then the following are equivalent: # has a closed graph (in ); # (definition) the graph of , , is a closed subset of ; # for every and net in such that in , if is such that the net in then ; #* Compare this to the definition of continuity in terms of nets, which recall is the following: for every and net in such that in , in . #* Thus to show that the function has a closed graph we ''may'' assume that converges in to some (and then show that ) while to show that is continuous we may ''not'' assume that converges in to some and we must instead prove that this is true (and moreover, we must more specifically prove that converges to in ). and if is a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

that is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, then we may add to this list:

is continuous;

and if both and are

first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...

spaces then we may add to this list:

has a sequentially closed graph (in );

;Function with a sequentially closed graph If is a function then the following are equivalent: # has a sequentially closed graph (in ); # (definition) the graph of is a sequentially closed subset of ; # for every and sequence in such that in , if is such that the net in then ; ;set-valued function with a closed graph If is a set-valued function between topological spaces and then the following are equivalent: # has a closed graph (in ); # (definition) the graph of is a closed subset of ; and if is compact and Hausdorff then we may add to this list:

is upper hemicontinuous and is a closed subset of for all ;

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

for all , , and sequences in and in such that in and in , and for all , then .

Characterizations of closed graphs (general topology)

Throughout, let

X

and

Y

be topological spaces and

X \times Y

is endowed with the product topology.

Function with a closed graph

f : X \to Y

is a function then it is said to have a if it satisfies any of the following are equivalent conditions:

(Definition): The graph $\operatorname f$ of $f$ is a closed subset of $X \times Y.$
For every $x \in X$ and net $x_ = \left(x_i\right)_$ in $X$ such that $x_ \to x$ in $X,$ if $y \in Y$ is such that the net $f\left(x_\right) = \left(f\left(x_i\right)\right)_ \to y$ in $Y$ then $y = f(x).$ * Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x \in X$ and net $x_ = \left(x_i\right)_$ in $X$ such that $x_ \to x$ in $X,$ $f\left(x_\right) \to f(x)$ in $Y.$ * Thus to show that the function $f$ has a closed graph, it ''may'' be assumed that $f\left(x_\right)$ converges in $Y$ to some $y \in Y$ (and then show that $y = f(x)$ ) while to show that $f$ is continuous, it may ''not'' be assumed that $f\left(x_\right)$ converges in $Y$ to some $y \in Y$ and instead, it must be proven that this is true (and moreover, it must more specifically be proven that $f\left(x_\right)$ converges to $f(x)$ in $Y$ ).

and if

Y

is a Hausdorff compact space then we may add to this list:

$f$ is continuous.

and if both

X

and

Y

are

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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
$x_ = \left(x_i\right)_^$ in $X$ such that $x_ \to x$ in $X,$ if $y \in Y$ is such that the net $f\left(x_\right) := \left(f\left(x_i\right)\right)_^ \to y$ in $Y$ then $y = f(x).$

Sufficient conditions for a closed graph

* If is a

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

between topological spaces and if is Hausdorff then has a closed graph in . However, if is a function between Hausdorff topological spaces, then it is possible for to have a closed graph in 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

where there are many theorems giving conditions under which a

with a closed graph is necessarily continuous. * If is a function between topological spaces whose graph is closed in and if is a

compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

then is continuous.

Examples

''For examples in functional analysis, see

continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...

.''

Continuous but ''not'' closed maps

* Let denote the real numbers with the usual

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...

and let denote with the

indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...

(where note that is ''not'' Hausdorff and that every function valued in is continuous). Let be defined by and for all . Then is continuous but its graph is ''not'' closed in . * If is any space then the identity map is continuous but its graph, which is the diagonal , is closed in if and only if is Hausdorff.Rudin p.50 In particular, if is not Hausdorff then is continuous but ''not'' closed. * If is a continuous map whose graph is not closed then is ''not'' a Hausdorff space.

Closed but ''not'' continuous maps

* Let and both denote the real numbers with the usual

. Let be defined by and for all . Then has a closed graph (and a sequentially closed graph) in but it is ''not'' continuous (since it has a discontinuity at ). * Let denote the real numbers with the usual

, let denote with the

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

, and let be the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

(i.e. for every ). Then is a

whose graph is closed in but it is clearly ''not'' continuous (since singleton sets are open in but not in ). * Let be a Hausdorff TVS and let be a vector topology on that is strictly finer than . Then the identity map is a closed discontinuous linear operator.

References

* * * * * * * * * * Functional analysis