Closed graph property

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

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 Template:Math is the set

Template:Math.

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

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

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

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

Template:Math.

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

• Note that in this case, Template:Math.

Open and closed graph

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

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

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

Note that we may define an open graph, a sequentially closed graph, and a sequentially open graph in similar ways.

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

Closable maps and closures

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

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

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

Closed maps and closed linear operators

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

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

Definition: A map Template:Math is called closed if its graph is closed in Template: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, "Template:Mvar is closed" may instead mean the following:

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

Function with a closed graph

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

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

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

Template:Mvar  is continuous;Template:Sfn

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

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

Function with a sequentially closed graph

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

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

set-valued function with a closed graph

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

1. Template:Mvar  has a closed graph (in Template:Math);
2. (definition) the graph of Template:Mvar is a closed subset of Template:Math;

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

Template:Mvar is upper hemicontinuous and Template:Math is a closed subset of Template:Mvar for all Template:Math;

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

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

Characterizations of closed graphs (general topology)

Throughout, let $X$ and $Y$ be topological spaces and $X\times Y$ is endowed with the product topology.

Function with a closed graph

If $f:X\to Y$ is a function then it is said to have a Template:Em if it satisfies any of the following are equivalent conditions:

1. (Definition): The graph $\mathrm{graph}f$ of $f$ is a closed subset of $X\times Y.$
2. For every $x\in X$ and net $x_{\bullet }={\left(x_i\right)}_{i\in I}$ in $X$ such that $x_{\bullet }\to x$ in $X,$ if $y\in Y$ is such that the net $f\left(x_{\bullet }\right)={\left(f\left(x_i\right)\right)}_{i\in I}\to y$ in $Y$ then $y=f(x).$Template:Sfn
   • Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x\in X$ and net $x_{\bullet }={\left(x_i\right)}_{i\in I}$ in $X$ such that $x_{\bullet }\to x$ in $X,$ $f\left(x_{\bullet }\right)\to f(x)$ in $Y.$
   • Thus to show that the function $f$ has a closed graph, it may be assumed that $f\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y$ (and then show that $y=f(x)$) while to show that $f$ is continuous, it may not be assumed that $f\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y$ and instead, it must be proven that this is true (and moreover, it must more specifically be proven that $f\left(x_{\bullet }\right)$ converges to $f(x)$ in $Y$).

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

3. $f$ is continuous.Template:Sfn

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

4. $f$ has a sequentially closed graph in $X\times Y.$

Function with a sequentially closed graph

If $f:X\to Y$ is a function then the following are equivalent:

1. $f$ has a sequentially closed graph in $X\times Y.$
2. Definition: the graph of $f$ is a sequentially closed subset of $X\times Y.$
3. For every $x\in X$ and sequence $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ in $X$ such that $x_{\bullet }\to x$ in $X,$ if $y\in Y$ is such that the net $f\left(x_{\bullet }\right):={\left(f\left(x_i\right)\right)}_{i=1}^{\mathrm{\infty }}\to y$ in $Y$ then $y=f(x).$Template:Sfn

Sufficient conditions for a closed graph

• If Template:Math is a continuous function between topological spaces and if Template:Mvar is Hausdorff then Template:Mvar  has a closed graph in Template:Math.Template:Sfn However, if Template:Math is a function between Hausdorff topological spaces, then it is possible for Template:Mvar  to have a closed graph in Template:Math but not be continuous.

Closed graph theorems: When a closed graph implies continuity

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

Examples

For examples in functional analysis, see continuous linear operator.

Continuous but not closed maps

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

Closed but not continuous maps

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

See also

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References

• Template:Köthe Topological Vector Spaces I
• Template:Kriegl Michor The Convenient Setting of Global Analysis
• Template:Munkres Topology
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Swartz An Introduction to Functional Analysis
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces