In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."
The graph of a function is the set
As usual, if and are topological spaces then is assumed to be endowed with the product topology.
If and are topological spaces, and is a function, then has a closed graph (resp. sequentially closed graph) in if the graph of is a closed (resp. sequentially closed) subset of If or if is clear from context then "in " may be omitted from writing.
A partial map , f : X ↣ Y, if a map from a subset of denoted by dom f, into If is written then it is meant that f : X ↣ Y is a partial map and dom f = D.denoted by
A map is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) if the graph of is closed (resp. sequentially closed) in (rather than in ).
A map is a linear or a linear operator if and are vector spaces, is a vector subspace of is a linear map.
Assume that and are topological vector spaces (TVSs).
A linear operator is called closed or a closed linear operator if its graph is closed in
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 ).
When reading literature in functional analysis, if is a linear map between topological vector spaces (TVSs) then " is closed" will almost always mean that its graph is closed. However, " is closed" may, especially in literature about point-set topology, instead mean the following:
A map between topological spaces is called a closed map 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.
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:
and if is a Hausdorff compact space then we may add to this list:
and if both and are first-countable spaces then we may add to this list:
If is a function then the following are equivalent:
Suppose is a linear operator between Banach spaces.
In functional analysis, the closed graph theorem states the following: If and are Banach spaces, and is a linear operator, then is continuous if and only if its graph is closed in (with the product topology).
The closed graph theorem can be reformulated may be rewritten into a form that is more easily usable:
Closed Graph Theorem for Banach spaces — If is a linear operator between Banach spaces, then the following are equivalent:
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.
The closed graph theorem can be generalized from Banach spaces to more abstract topological vector spaces in the following ways.
There are versions that does not require to be locally convex.
Theorem — A linear map between two F-spaces is continuous if and only if its graph is closed.
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:
Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.
An even more general version of the closed graph theorem is
Under this condition, if is a linear map whose graph is closed 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.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 u is a Borel set in then u is continuous.
An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces.
A topological space is called a Kσδ 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 Kσδ space (that is, if there is a Kσδ 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:
If is closed linear operator from a Hausdorff locally convex TVS into a Hausdorff finite-dimensional TVS then F is continuous.
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