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 ...
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
and
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
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
of
where
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
is said to have a
(in
) if its graph
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 ...
(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,
is said to have a if
is a
sequentially closed subset of
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
A partial function
is declared with the notation
which indicates that
has prototype
(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
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
) and that
is a dense subset of
Since the domain is denoted by
it is not always necessary to assign a symbol (such as
) to a partial function's domain, in which case the notation
or
may be used to indicate that
is a partial function with codomain
whose domain
is a dense subset of
A densely defined linear operator between vector spaces is a partial function
whose domain
is a dense vector subspace of a TVS
such that
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
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
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
is (as before) the set
However, one exception to this is the definition of "closed graph". A function
is said to have a closed graph (respectively, a sequentially closed graph) if
is a closed (respectively, sequentially closed) subset of
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
and
as it was defined above for ordinary functions.
[In contrast, when is considered as an ordinary function (rather than as the partial function ), then "having a closed graph" would instead mean that is a closed subset of If is a closed subset of then it is also a closed subset of although the converse is not guaranteed in general.]
Closable maps and closures
A linear operator
is in
if there exists a
containing
and a function (resp. multifunction)
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
If
is a closable linear operator then a or an 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
).
Characterizations of closed graphs (general topology)
Throughout, let
and
be topological spaces and
is endowed with the product topology.
Function with a closed graph
If
is a function then it is said to have a if it satisfies any of the following are equivalent conditions:
- (Definition): The graph of is a closed subset of
- For every 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 ...
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, it may be assumed that converges in to some (and then show that ) while to show that is continuous, it may not be assumed that converges in to some and instead, it must be proven that this is true (and moreover, it must more specifically be proven that converges to in ).
and if
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:
- 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 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 ...
in such that in if is such that the net in then
Basic properties of maps with closed graphs
Suppose
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.
- If is closed then is closed where is a scalar and is the
identity function
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, un ...
.
- If is closed, then its
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
(or nullspace) is a closed vector subspace of
- If is closed and
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
then its inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
is also closed.
- A linear operator admits a closure if and only if for every and every pair of sequences and in both converging to in such that both and converge in one has
Examples and counterexamples
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
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 is 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. 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
-
If is a Hausdorff TVS and is a vector topology on that is strictly finer than then the identity map a closed discontinuous linear operator.
-
Consider 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 where is the Banach space of all continuous function
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 val ...
s on an interval
If one takes its domain to be then is a closed operator, which is not bounded.
On the other hand, if is the space of smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s scalar valued functions then will no longer be closed, but it will be closable, with the closure being its extension defined on
-
Let and both 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
. 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
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
, 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 top ...
, 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 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 ...
whose graph is closed in but it is clearly not continuous (since singleton sets are open in but not in ).
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
* ...
of
is
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
whose domain is a strict subset of
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
and
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
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
is called a
if it is the countable intersection of countable unions of compact sets.
A Hausdorff topological space
is called K-analytic if it is the continuous image of a
space (that is, if there is a
space
and a continuous map of
onto
).
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
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
into a Hausdorff finite-dimensional TVS
then
is continuous.
See also
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References
Notes
Bibliography
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{{TopologicalVectorSpaces
Theorems in functional analysis