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In mathematics, 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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their
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 discre ...
. In its most elementary form, the closed graph theorem states that a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
between two
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
is continuous
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the graph of that function is closed. The closed graph theorem has extensive application throughout functional analysis, because it can control whether a partially-defined linear operator admits continuous extensions. For this reason, it has been generalized to many circumstances beyond the elementary formulation above.


Preliminaries

The closed graph theorem is a result about
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 ...
f : X \to Y between two
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
endowed with
topologies In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
making them into
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 ...
s (TVSs). We will henceforth assume that X and Y are topological vector spaces, such as
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
s for example, and that
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
s, such as X \times Y, are 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-seem ...
. The of this function is the subset \operatorname = \, of \operatorname (f) \times Y = X \times Y, where \operatorname f = X denotes the function's
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
. The map f : X \to Y is said to have a (in X \times Y) if its graph \operatorname f 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 ...
of
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 ...
X \times Y (with the usual
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-seem ...
). Similarly, f is said to have a if \operatorname f is a sequentially closed subset of X \times Y. A is a linear map whose graph is closed (it need not be continuous or bounded). It is common in functional analysis to call such maps "closed", but this should not be confused the non-equivalent notion of 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, ...
" that appears in
general topology In mathematics, general 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 differential topology, geometri ...
. Partial functions It is common in functional analysis to consider
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
s, which are functions defined on a dense
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of some space X. A partial function f is declared with the notation f : D \subseteq X \to Y, which indicates that f has prototype f : D \to Y (that is, its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
is D and its
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
is Y) and that \operatorname f = D is a dense subset of X. Since the domain is denoted by \operatorname f, it is not always necessary to assign a symbol (such as D) to a partial function's domain, in which case the notation f : X \rightarrowtail Y or f : X \rightharpoonup Y may be used to indicate that f is a partial function with codomain Y whose domain \operatorname f is a dense subset of X. A densely defined linear operator between vector spaces is a partial function f : D \subseteq X \to Y whose domain D is a dense vector subspace of a TVS X such that f : D \to Y 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 ...
. A prototypical example of a partial function is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
operator, which is only defined on the space D := C^1(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
of once
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
functions, a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of the space X := C(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
of
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
. Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the
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 discre ...
of a partial function f is (as before) the set \operatorname = \. However, one exception to this is the definition of "closed graph". A function f : D \subseteq X \to Y is said to have a closed graph (respectively, a sequentially closed graph) if \operatorname f is a closed (respectively, sequentially closed) subset of X \times Y in 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-seem ...
; importantly, note that the product space is X \times Y and D \times Y = \operatorname f \times Y as it was defined above for ordinary functions.In contrast, when f : D \to Y is considered as an ordinary function (rather than as the partial function f : D \subseteq X \to Y), then "having a closed graph" would instead mean that \operatorname f is a closed subset of D \times Y. If \operatorname f is a closed subset of X \times Y then it is also a closed subset of \operatorname (f) \times Y although the converse is not guaranteed in general.


Closable maps and closures

A linear operator f : D \subseteq X \to Y is in X \times Y if there exists a E \subseteq X containing D and a function (resp. multifunction) F : E \to Y whose graph is equal to the closure of the set \operatorname f in X \times Y. Such an F is called a closure of f in X \times Y, is denoted by \overline, and necessarily extends f. If f : D \subseteq X \to Y is a closable linear operator then a or an of f is a subset C \subseteq D such that the closure in X \times Y of the graph of the restriction f\big\vert_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 \operatorname f in X \times Y is equal to the closure of \operatorname f\big\vert_C in X \times Y).


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

If f : X \to Y is a function then it is said to have a if it satisfies any of the following are equivalent conditions:
  1. (Definition): The graph \operatorname f of f is a closed subset of X \times Y.
  2. For every x \in X and
    net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
    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 In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
then we may add to this list:
  1. f is continuous.
and if both X and Y 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:
  1. 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:
  1. f has a sequentially closed graph in X \times Y.
  2. Definition: the graph of f is a sequentially closed subset of X \times Y.
  3. 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 called ...
    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).


Basic properties of maps with closed graphs

Suppose f : D(f) \subseteq X \to Y is a linear operator between
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
s.


Examples and counterexamples


Continuous but not closed maps


Closed but not continuous maps


Closed graph theorems


Between Banach spaces

The operator is required to be everywhere-defined, that is, the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
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(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
, whose domain is a strict subset of C(
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
. 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.


Complete metrizable codomain

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


Between F-spaces

There are versions that does not require Y to be locally convex. This theorem is restated and extend it with some conditions that can be used to determine if a graph is closed:


Complete pseudometrizable codomain

Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.


Codomain not complete or (pseudo) metrizable

An even more general version of the closed graph theorem is


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. Recall that a topological space is called a
Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
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: An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces. A topological space X is called a K_ if it is the countable intersection of countable unions of compact sets. A Hausdorff topological space Y is called K-analytic if it is the continuous image of a K_ space (that is, if there is a K_ 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
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
is K-analytic as is the weak dual of a Frechet space. The generalized Borel graph theorem states:


Related results

If F : X \to Y is closed linear operator from a Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
TVS X into a Hausdorff finite-dimensional TVS Y then F is continuous.


See also

* * * * * * * * * *


References

Notes


Bibliography

* * * * * * * * * * * * * * * * * * * * * * {{TopologicalVectorSpaces Theorems in functional analysis