Closed graph theorem (functional analysis)

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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."

Contents

Definitions

Graphs and closed graphs

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.

Linear operators

A partial map , [1] denoted by f : XY, if a map from a subset of denoted by dom f, into If is written then it is meant that f : XY is a partial map and dom f = D.

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.

Closed linear operators

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

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 ).

Closed maps vs. closed linear operators

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.

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:

  1. (Definition): The graph of is a closed subset of
  2. 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 ).

and if is a Hausdorff compact space then we may add to this list:

  1. is continuous. [3]

and if both and are first-countable spaces then we may add to this list:

  1. has a sequentially closed graph in
Function with a sequentially closed graph

If is a function then the following are equivalent:

  1. has a sequentially closed graph in
  2. Definition: the graph of is a sequentially closed subset of
  3. 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.

Examples and counterexamples

Continuous but not closed maps

Closed but not continuous maps

Closed graph theorems

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:

  1. is continuous.
  2. is a closed operator (that is, the graph of is closed).
  3. If in then in
  4. If in then in
  5. If in and if converges in to some then
  6. 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.

Theorem  A linear map between two F-spaces is continuous if and only if its graph is closed. [7] [8]

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:

  1. is continuous.
  2. has a closed graph.
  3. If in and if converges in to some then [9]
  4. 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 u is a Borel set in then u is continuous. [13]

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:

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 u is closed in then u is continuous.

If is closed linear operator from a Hausdorff locally convex TVS into a Hausdorff finite-dimensional TVS then F is continuous. [15]

See also

Related Research Articles

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References

  1. Dolecki & Mynard 2016, pp. 4-5.
  2. 1 2 3 4 5 Narici & Beckenstein 2011, pp. 459-483.
  3. Munkres 2000, p. 171.
  4. Rudin 1991, p. 50.
  5. Narici & Beckenstein 2011, p. 480.
  6. Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN   0-471-50731-8.
  7. Schaefer & Wolff 1999, p. 78.
  8. Trèves (1995) , p. 173
  9. Rudin 1991, pp. 50-52.
  10. 1 2 3 Narici & Beckenstein 2011, pp. 474-476.
  11. Narici & Beckenstein 2011, p. 479-483.
  12. Trèves 2006, p. 169.
  13. 1 2 Trèves 2006, p. 549.
  14. Trèves 2006, pp. 557-558.
  15. Narici & Beckenstein 2011, p. 476.

Bibliography