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.
Aclosed 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 densesubset of some space
A partial function is declared with the notation which indicates that has prototype (that is, its domainis and its codomainis) 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]Adensely 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 ofcontinuous 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]
A linear operator isclosablein 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 domainof is a subset such that the closure in of the graph of the restriction ofto 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)[edit]
Throughout, let and be topological spaces and is endowed with the product topology.
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 netin such that inif 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 inin
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).
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
Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology (where isnot 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 isnot a Hausdorff space.
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 ofsmooth 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 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 topologyon
Ifin then in
Ifin then in
Ifin and if converges in to some then
Ifin and if converges in to some then
The operator is required to be everywhere-defined, that is, the domainofis 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.
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[edit]
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.
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 convexHausdorff 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 reflexiveFréchet 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.
^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.
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Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN978-0-387-90081-0. OCLC878109401.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN978-3-642-64988-2. MR0248498. OCLC840293704.