Английская Википедия:Closed graph property

Шаблон:Short description Шаблон:Redirect In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.^[1]^[2] A function Шаблон:Math between topological spaces has a closed graph if its graph is a closed subset of the product space Шаблон:Math. A related property is open graph.^[3]

This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

Definitions

Graphs and set-valued functions

Definition and notation: The graph of a function Шаблон:Math is the set

Шаблон:Math.

Notation: If Шаблон:Mvar is a set then the power set of Шаблон:Mvar, which is the set of all subsets of Шаблон:Mvar, is denoted by Шаблон:Math or Шаблон:Math.

Definition: If Шаблон:Mvar and Шаблон:Mvar are sets, a set-valued function in Шаблон:Mvar on Шаблон:Mvar (also called a Шаблон:Mvar-valued multifunction on Шаблон:Mvar) is a function Шаблон:Math with domain Шаблон:Mvar that is valued in Шаблон:Math. That is, Шаблон:Mvar is a function on Шаблон:Mvar such that for every Шаблон:Math, Шаблон:Math is a subset of Шаблон:Mvar.

Some authors call a function Шаблон:Math a set-valued function only if it satisfies the additional requirement that Шаблон:Math is not empty for every Шаблон:Math; this article does not require this.

Definition and notation: If Шаблон:Math is a set-valued function in a set Шаблон:Mvar then the graph of Шаблон:Mvar is the set

Шаблон:Math.

Definition: A function Шаблон:Math can be canonically identified with the set-valued function Шаблон:Math defined by Шаблон:Math for every Шаблон:Math, where Шаблон:Mvar is called the canonical set-valued function induced by (or associated with) Шаблон:Mvar.

Note that in this case, Шаблон:Math.

Open and closed graph

We give the more general definition of when a Шаблон:Mvar-valued function or set-valued function defined on a subset Шаблон:Mvar of Шаблон:Mvar has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace Шаблон:Mvar of a topological vector space Шаблон:Mvar (and not necessarily defined on all of Шаблон:Mvar). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

Assumptions: Throughout, Шаблон:Mvar and Шаблон:Mvar are topological spaces, Шаблон:Math, and Шаблон:Mvar is a Шаблон:Mvar-valued function or set-valued function on Шаблон:Mvar (i.e. Шаблон:Math or Шаблон:Math). Шаблон:Math will always be endowed with the product topology.

Definition:Шаблон:Sfn We say that Шаблон:Mvar has a closed graph (resp. open graph, sequentially closed graph, sequentially open graph) in Шаблон:Math if the graph of Шаблон:Mvar, Шаблон:Math, is a closed (resp. open, sequentially closed, sequentially open) subset of Шаблон:Math when Шаблон:Math is endowed with the product topology. If Шаблон:Math or if Шаблон:Mvar is clear from context then we may omit writing "in Шаблон:Math"

Observation: If Шаблон:Math is a function and Шаблон:Mvar is the canonical set-valued function induced by Шаблон:Mvar (i.e. Шаблон:Math is defined by Шаблон:Math for every Шаблон:Math) then since Шаблон:Math, Шаблон:Mvar has a closed (resp. sequentially closed, open, sequentially open) graph in Шаблон:Math if and only if the same is true of Шаблон:Mvar.

Closable maps and closures

Definition: We say that the function (resp. set-valued function) Шаблон:Mvar is closable in Шаблон:Math if there exists a subset Шаблон:Math containing Шаблон:Mvar and a function (resp. set-valued function) Шаблон:Math whose graph is equal to the closure of the set Шаблон:Math in Шаблон:Math. Such an Шаблон:Mvar is called a closure of Шаблон:Mvar in Шаблон:Math, is denoted by Шаблон:Math, and necessarily extends Шаблон:Mvar.

Additional assumptions for linear maps: If in addition, Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar are topological vector spaces and Шаблон:Math is a linear map then to call Шаблон:Mvar closable we also require that the set Шаблон:Mvar be a vector subspace of Шаблон:Mvar and the closure of Шаблон:Mvar be a linear map.

Definition: If Шаблон:Mvar is closable on Шаблон:Mvar then a core or essential domain of Шаблон:Mvar is a subset Шаблон:Math such that the closure in Шаблон:Math of the graph of the restriction Шаблон:Math of Шаблон:Mvar to Шаблон:Mvar is equal to the closure of the graph of Шаблон:Mvar in Шаблон:Math (i.e. the closure of Шаблон:Math in Шаблон:Math is equal to the closure of Шаблон:Math in Шаблон:Math).

Closed maps and closed linear operators

Definition and notation: When we write Шаблон:Math then we mean that Шаблон:Mvar is a Шаблон:Mvar-valued function with domain Шаблон:Math where Шаблон:Math. If we say that Шаблон:Math is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of Шаблон:Mvar is closed (resp. sequentially closed) in Шаблон:Math (rather than in Шаблон:Math).

When reading literature in functional analysis, if Шаблон:Math is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "Шаблон:Mvar is closed" will almost always means the following:

Definition: A map Шаблон:Math is called closed if its graph is closed in Шаблон:Math. In particular, the term "closed linear operator" will almost certainly refer to a linear map whose graph is closed.

Otherwise, especially in literature about point-set topology, "Шаблон:Mvar is closed" may instead mean the following:

Definition: A map Шаблон:Math between topological spaces is called a closed map if the image of a closed subset of Шаблон:Mvar is a closed subset of Шаблон:Mvar.

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

Throughout, let Шаблон:Mvar and Шаблон:Mvar be topological spaces.

Function with a closed graph

If Шаблон:Math is a function then the following are equivalent:

Шаблон:Mvar has a closed graph (in Шаблон:Math);
(definition) the graph of Шаблон:Mvar, Шаблон:Math, is a closed subset of Шаблон:Math;
for every Шаблон:Math and net Шаблон:Math in Шаблон:Mvar such that Шаблон:Math in Шаблон:Mvar, if Шаблон:Math is such that the net Шаблон:Math in Шаблон:Mvar then Шаблон:Math;Шаблон:Sfn
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every Шаблон:Math and net Шаблон:Math in Шаблон:Mvar such that Шаблон:Math in Шаблон:Mvar, Шаблон:Math in Шаблон:Mvar.
- Thus to show that the function Шаблон:Mvar has a closed graph we may assume that Шаблон:Math converges in Шаблон:Mvar to some Шаблон:Math (and then show that Шаблон:Math) while to show that Шаблон:Mvar is continuous we may not assume that Шаблон:Math converges in Шаблон:Mvar to some Шаблон:Math and we must instead prove that this is true (and moreover, we must more specifically prove that Шаблон:Math converges to Шаблон:Math in Шаблон:Mvar).

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

Шаблон:Mvar is continuous;Шаблон:Sfn

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

Шаблон:Mvar has a sequentially closed graph (in Шаблон:Math);

Function with a sequentially closed graph

If Шаблон:Math is a function then the following are equivalent:

Шаблон:Mvar has a sequentially closed graph (in Шаблон:Math);
(definition) the graph of Шаблон:Mvar is a sequentially closed subset of Шаблон:Math;
for every Шаблон:Math and sequence Шаблон:Math in Шаблон:Mvar such that Шаблон:Math in Шаблон:Mvar, if Шаблон:Math is such that the net Шаблон:Math in Шаблон:Mvar then Шаблон:Math;Шаблон:Sfn

set-valued function with a closed graph

If Шаблон:Math is a set-valued function between topological spaces Шаблон:Mvar and Шаблон:Mvar then the following are equivalent:

Шаблон:Mvar has a closed graph (in Шаблон:Math);
(definition) the graph of Шаблон:Mvar is a closed subset of Шаблон:Math;

and if Шаблон:Mvar is compact and Hausdorff then we may add to this list:

Шаблон:Mvar is upper hemicontinuous and Шаблон:Math is a closed subset of Шаблон:Mvar for all Шаблон:Math;^[4]

and if both Шаблон:Mvar and Шаблон:Mvar are metrizable spaces then we may add to this list:

for all Шаблон:Math, Шаблон:Math, and sequences Шаблон:Math in Шаблон:Mvar and Шаблон:Math in Шаблон:Mvar such that Шаблон:Math in Шаблон:Mvar and Шаблон:Math in Шаблон:Mvar, and Шаблон:Math for all Шаблон:Mvar, then Шаблон:Math.Шаблон:Citation needed

Sufficient conditions for a closed graph

If Шаблон:Math is a continuous function between topological spaces and if Шаблон:Mvar is Hausdorff then Шаблон:Mvar has a closed graph in Шаблон:Math.Шаблон:Sfn
- Note that if Шаблон:Math is a function between Hausdorff topological spaces then it is possible for Шаблон:Mvar to have a closed graph in Шаблон:Math but not be continuous.

Closed graph theorems: When a closed graph implies continuity

Шаблон:Main

Conditions that guarantee that a function with a closed graph is necessarily continuous are called closed graph theorems. Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous.

If Шаблон:Math is a function between topological spaces whose graph is closed in Шаблон:Math and if Шаблон:Mvar is a compact space then Шаблон:Math is continuous.Шаблон:Sfn

Examples

Continuous but not closed maps

Let Шаблон:Mvar denote the real numbers Шаблон:Math with the usual Euclidean topology and let Шаблон:Mvar denote Шаблон:Math with the indiscrete topology (where note that Шаблон:Mvar is not Hausdorff and that every function valued in Шаблон:Mvar is continuous). Let Шаблон:Math be defined by Шаблон:Math and Шаблон:Math for all Шаблон:Math. Then Шаблон:Math is continuous but its graph is not closed in Шаблон:Math.Шаблон:Sfn
If Шаблон:Mvar is any space then the identity map Шаблон:Math is continuous but its graph, which is the diagonal Шаблон:Math, is closed in Шаблон:Math if and only if Шаблон:Mvar is Hausdorff.^[5] In particular, if Шаблон:Mvar is not Hausdorff then Шаблон:Math is continuous but not closed.
If Шаблон:Math is a continuous map whose graph is not closed then Шаблон:Mvar is not a Hausdorff space.

Closed but not continuous maps

Let Шаблон:Mvar and Шаблон:Mvar both denote the real numbers Шаблон:Math with the usual Euclidean topology. Let Шаблон:Math be defined by Шаблон:Math and Шаблон:Math for all Шаблон:Math. Then Шаблон:Math has a closed graph (and a sequentially closed graph) in Шаблон:Math but it is not continuous (since it has a discontinuity at Шаблон:Math).Шаблон:Sfn
Let Шаблон:Mvar denote the real numbers Шаблон:Math with the usual Euclidean topology, let Шаблон:Mvar denote Шаблон:Math with the discrete topology, and let Шаблон:Math be the identity map (i.e. Шаблон:Math for every Шаблон:Math). Then Шаблон:Math is a linear map whose graph is closed in Шаблон:Math but it is clearly not continuous (since singleton sets are open in Шаблон:Mvar but not in Шаблон:Mvar).Шаблон:Sfn
Let Шаблон:Math be a Hausdorff TVS and let Шаблон:Math be a vector topology on Шаблон:Mvar that is strictly finer than Шаблон:Math. Then the identity map Шаблон:Math a closed discontinuous linear operator.Шаблон:Sfn

Closed linear operators

Шаблон:See also

Every continuous linear operator valued in a Hausdorff topological vector space (TVS) has a closed graph and recall that a linear operator between two normed spaces is continuous if and only if it is bounded.

Definition: If Шаблон:Mvar and Шаблон:Mvar are topological vector spaces (TVSs) then we call a linear map Шаблон:Math a closed linear operator if its graph is closed in Шаблон:Math.

Closed graph theorem

The closed graph theorem states that any closed linear operator Шаблон:Math between two F-spaces (such as Banach spaces) is continuous, where recall that if Шаблон:Mvar and Шаблон:Mvar are Banach spaces then Шаблон:Math being continuous is equivalent to Шаблон:Mvar being bounded.

Basic properties

The following properties are easily checked for a linear operator Шаблон:Math between Banach spaces:

If Шаблон:Mvar is closed then Шаблон:Math is closed where Шаблон:Mvar is a scalar and Шаблон:Math is the identity function;
If Шаблон:Mvar is closed, then its kernel (or nullspace) is a closed vector subspace of Шаблон:Mvar;
If Шаблон:Mvar is closed and injective then its inverse Шаблон:Math is also closed;
A linear operator Шаблон:Mvar admits a closure if and only if for every Шаблон:Math and every pair of sequences Шаблон:Math and Шаблон:Math in Шаблон:Math both converging to Шаблон:Mvar in Шаблон:Mvar, such that both Шаблон:Math and Шаблон:Math converge in Шаблон:Mvar, one has Шаблон:Math.

Example

Consider the derivative operator Шаблон:Math where Шаблон:Math is the Banach space of all continuous functions on an interval Шаблон:Math. If one takes its domain Шаблон:Math to be Шаблон:Math, then Шаблон:Mvar is a closed operator, which is not bounded.^[6] On the other hand if Шаблон:Math, then Шаблон:Mvar will no longer be closed, but it will be closable, with the closure being its extension defined on Шаблон:Math.

References

Шаблон:Reflist

[1] Шаблон:Cite journal

[2] Шаблон:Cite journal

[:2-3] Шаблон:Cite journal

[aliprantis-4] Шаблон:Cite book

[5] Rudin p.50

[6] Шаблон:Cite book

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