In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the Closed graph property holds.
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 between two vector spaces endowed with topologies making them into topological vector spaces (TVSs). We will henceforth assume that
and
are topological vector spaces, such as Banach spaces for example, and that Cartesian products, such as
are endowed with the product topology. The graph of this function is the subset
A closed linear operator 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" that appears in general topology.
Partial functions
It is common in functional analysis to consider partial functions, which are functions defined on a dense subset of some space A partial function
is declared with the notation
which indicates that
has prototype
(that is, its domain is
and its codomain 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
[1] 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. A prototypical example of a partial function is the derivative operator, which is only defined on the space
of once continuously differentiable functions, a dense subset of the space
of continuous functions.
Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function is (as before) the set
However, one exception to this is the definition of "closed graph". A partial 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; importantly, note that the product space is
and not
as it was defined above for ordinary functions.[note 1]
Closable maps and closures
A linear operator is closable in
if there exists a vector subspace
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 core or an 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
).
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 closed graph if it satisfies any of the following are equivalent conditions:
- (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
[2]
- 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
).
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every
and if is a Hausdorff compact space then we may add to this list:
is continuous.[3]
and if both and
are first-countable 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
[2]
Basic properties of maps with closed graphs
Suppose is a linear operator between Banach spaces.
- If
is closed then
is closed where
is a scalar and
is the identity function.
- If
is closed, then its kernel (or nullspace) is a closed vector subspace of
- If
is closed and injective then its inverse
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 and let
denote
with the indiscrete topology (where
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
[2]
- 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.[4] 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.[5]
- Consider the derivative operator
where
is the Banach space of all continuous functions on an interval
If one takes its domain
to be
then
is a closed operator, which is not bounded.[6] On the other hand, if
is the space
of smooth functions 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. 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
).[2]
- Let
denote the real numbers
with the usual Euclidean topology, let
denote
with the discrete topology, and let
be the identity map (i.e.
for every
). Then
is a linear map whose graph is closed in
but it is clearly not continuous (since singleton sets are open in
but not in
).[2]
Closed graph theorems
Between Banach spaces
Closed Graph Theorem for Banach spaces — If is an everywhere-defined linear operator between Banach spaces, then the following are equivalent:
is continuous.
is closed (that is, the graph of
is closed in the product topology on
- If
in
then
in
- If
in
then
in
- If
in
and if
converges in
to some
then
- If
in
and if
converges in
to some
then
The operator is required to be everywhere-defined, that is, the domain 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 the following ways.
Theorem — A linear operator from a barrelled space to a Fréchet space
is continuous if and only if its graph is closed.
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:
Theorem — If is a linear map between two F-spaces, then the following are equivalent:
is continuous.
has a closed graph.
- If
in
and if
converges in
to some
then
[9]
- If
in
and if
converges in
to some
then
Complete pseudometrizable codomain
Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.
Closed Graph Theorem[10] — Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.
Closed Graph Theorem — A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous.[10]
Codomain not complete or (pseudo) metrizable
Theorem[11] — Suppose that is a linear map whose graph is closed. If
is an inductive limit of Baire TVSs and
is a webbed space then
is continuous.
Closed Graph Theorem[10] — A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.
An even more general version of the closed graph theorem is
Theorem[12] — Suppose that and
are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:
- If
is any closed subspace of
and
is any continuous map of
onto
then
is an open mapping.
Under this condition, if is a linear map whose graph is closed then
is continuous.
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.[13] Recall that a topological space is called a Polish space 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:
Borel Graph Theorem — Let be linear map between two locally convex Hausdorff spaces
and
If
is the inductive limit of an arbitrary family of Banach spaces, if
is a Souslin space, and if the graph of
is a Borel set in
then
is continuous.[13]
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 is K-analytic as is the weak dual of a Frechet space. The generalized Borel graph theorem states:
Generalized Borel Graph Theorem[14] — Let be a linear map between two locally convex Hausdorff spaces
and
If
is the inductive limit of an arbitrary family of Banach spaces, if
is a K-analytic space, and if the graph of
is closed in
then
is continuous.
Related results
If is closed linear operator from a Hausdorff locally convex TVS
into a Hausdorff finite-dimensional TVS
then
is continuous.[15]
See also
- Almost open linear map – Map that satisfies a condition similar to that of being an open map.
- Barrelled space – Type of topological vector space
- Closed graph – Graph of a map closed in the product space
- Closed linear operator – Graph of a map closed in the product space
- Densely defined operator – Function that is defined almost everywhere (mathematics)
- Discontinuous linear map
- Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions
- Open mapping theorem (functional analysis) – Condition for a linear operator to be open
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
- Webbed space – Space where open mapping and closed graph theorems hold
References
Notes
Bibliography
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