Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function $f : X → Y$ between topological spaces has a closed graph if its graph is a closed subset of the product space $X ×&thinsp;Y$. 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.

Graphs and set-valued functions

 * Definition and notation: The graph of a function $f : X → Y$ is the set


 * Notation: If $Y$ is a set then the power set of $Y$, which is the set of all subsets of $Y$, is denoted by $Gr f := { (x, f(x)) : x ∈ X&thinsp;} = { (x, y) ∈ X ×&thinsp;Y : y = f(x)&thinsp;}$ or $2^{Y}$.


 * Definition: If $X$ and $Y$ are sets, a set-valued function in $Y$ on $X$ (also called a $Y$-valued multifunction on $X$) is a function $𝒫(Y)$ with domain $X$ that is valued in $F : X → 2^{Y}$. That is, $F$ is a function on $X$ such that for every $2^{Y}$, $x ∈ X$ is a subset of $Y$.
 * Some authors call a function $F(x)$ a set-valued function only if it satisfies the additional requirement that $F : X → 2^{Y}$ is not empty for every $F(x)$; this article does not require this.


 * Definition and notation: If $x ∈ X$ is a set-valued function in a set $Y$ then the graph of $F$ is the set


 * Definition: A function $F : X → 2^{Y}$ can be canonically identified with the set-valued function $Gr F := { (x, y) ∈ X ×&thinsp;Y : y ∈ F(x)&thinsp;}$ defined by $f : X → Y$ for every $F : X → 2^{Y}$, where $F$ is called the canonical set-valued function induced by (or associated with) $f$.
 * Note that in this case, $F(x) := { f(x)&thinsp;}$.

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


 * Assumptions: Throughout, $X$ and $Y$ are topological spaces, $x ∈ X$, and $f$ is a $Y$-valued function or set-valued function on $S$ (i.e. $Gr f = Gr F$ or $S ⊆ X$). $f : S → Y$ will always be endowed with the product topology.


 * Definition: We say that $f$&thinsp; has a closed graph (resp. open graph, sequentially closed graph, sequentially open graph) in $f : S → 2^{Y}$ if the graph of $f$, $X ×&thinsp;Y$, is a closed (resp. open, sequentially closed, sequentially open) subset of $X ×&thinsp;Y$ when $Gr f$ is endowed with the product topology. If $X ×&thinsp;Y$ or if $X$ is clear from context then we may omit writing "in $X ×&thinsp;Y$"


 * Observation: If $S = X$ is a function and $G$ is the canonical set-valued function induced by $g$&thinsp; (i.e. $X ×&thinsp;Y$ is defined by $g : S → Y$ for every $G : S → 2^{Y}$) then since $G(s) := { g(s)&thinsp;}$, $g$ has a closed (resp. sequentially closed, open, sequentially open) graph in $s ∈ S$ if and only if the same is true of $G$.

Closable maps and closures

 * Definition: We say that the function (resp. set-valued function) $f$ is closable in $Gr g = Gr G$ if there exists a subset $X ×&thinsp;Y$ containing $S$ and a function (resp. set-valued function) $X ×&thinsp;Y$ whose graph is equal to the closure of the set $D ⊆ X$ in $F : D → Y$. Such an $F$ is called a closure of $f$ in $Gr f$, is denoted by $X ×&thinsp;Y$, and necessarily extends $f$.
 * Additional assumptions for linear maps: If in addition, $S$, $X$, and $Y$ are topological vector spaces and $X ×&thinsp;Y$ is a linear map then to call $f$ closable we also require that the set $D$ be a vector subspace of $X$ and the closure of $f$ be a linear map.


 * Definition: If $f$ is closable on $S$ then a core or essential domain of $f$ is a subset $\overline{f}$ such that the closure in $f : S → Y$ of the graph of the restriction $D ⊆ S$ of $f$ to $D$ is equal to the closure of the graph of $f$ in $X ×&thinsp;Y$ (i.e. the closure of $f |_{D} : D → Y$ in $X ×&thinsp;Y$ is equal to the closure of $Gr f$ in $X ×&thinsp;Y$).

Closed maps and closed linear operators

 * Definition and notation: When we write $Gr f |_{D}$ then we mean that $f$ is a $Y$-valued function with domain $X ×&thinsp;Y$ where $f : D(f) ⊆ X → Y$. If we say that $D(f)$ is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of $f$ is closed (resp. sequentially closed) in $D(f) ⊆ X$ (rather than in $f : D(f) ⊆ X → Y$).

When reading literature in functional analysis, if $X ×&thinsp;Y$ is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "$f$ is closed" will almost always means the following:


 * Definition: A map $D(f) ×&thinsp;Y$ is called closed if its graph is closed in $f : X → Y$. 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, "$f$ is closed" may instead mean the following:


 * Definition: A map $f : X → Y$ between topological spaces is called a closed map if the image of a closed subset of $X$ is a closed subset of $Y$.

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 $X$ and $Y$ be topological spaces.


 * Function with a closed graph

If $X ×&thinsp;Y$ is a function then the following are equivalent:


 * 1) $f$&thinsp; has a closed graph (in $f : X → Y$);
 * 2) (definition) the graph of $f$, $f : X → Y$, is a closed subset of $X ×&thinsp;Y$;
 * 3) for every $Gr f$ and net $X ×&thinsp;Y$ in $X$ such that $x ∈ X$ in $X$, if $x_{•} = (x_{i})_{i ∈ I}$ is such that the net $x_{•} → x$ in $Y$ then $y ∈ Y$;
 * 4) * Compare this to the definition of continuity in terms of nets, which recall is the following: for every $f(x_{•}) := (f(x_{i}))_{i ∈ I} → y$ and net $y = f(x)$ in $X$ such that $x ∈ X$ in $X$, $x_{•} = (x_{i})_{i ∈ I}$ in $Y$.
 * 5) * Thus to show that the function $f$ has a closed graph we may assume that $x_{•} → x$ converges in $Y$ to some $f(x_{•}) → f(x)$ (and then show that $f(x_{•})$) while to show that $f$ is continuous we may not assume that $y ∈ Y$ converges in $Y$ to some $y = f(x)$ and we must instead prove that this is true (and moreover, we must more specifically prove that $f(x_{•})$ converges to $y ∈ Y$ in $Y$).

and if $Y$ is a Hausdorff compact space then we may add to this list:
 * 1) $f$&thinsp; is continuous;

and if both $X$ and $Y$ are first-countable spaces then we may add to this list:
 * 1) $f$&thinsp; has a sequentially closed graph (in $f(x_{•})$);


 * Function with a sequentially closed graph

If $f(x)$ is a function then the following are equivalent:
 * 1) $f$&thinsp; has a sequentially closed graph (in $X ×&thinsp;Y$);
 * 2) (definition) the graph of $f$ is a sequentially closed subset of $f : X → Y$;
 * 3) for every $X ×&thinsp;Y$ and sequence $X ×&thinsp;Y$ in $X$ such that $x ∈ X$ in $X$, if $x_{•} = (x_{i})∞ i=1$ is such that the net $x_{•} → x$ in $Y$ then $y ∈ Y$;


 * set-valued function with a closed graph

If $f(x_{•}) := (f(x_{i}))∞ i=1 → y$ is a set-valued function between topological spaces $X$ and $Y$ then the following are equivalent:
 * 1) $F$&thinsp; has a closed graph (in $y = f(x)$);
 * 2) (definition) the graph of $F$ is a closed subset of $F : X → 2^{Y}$;

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


 * 1) $F$ is upper hemicontinuous and $X ×&thinsp;Y$ is a closed subset of $Y$ for all $X ×&thinsp;Y$; 

and if both $X$ and $Y$ are metrizable spaces then we may add to this list:
 * 1) for all $F(x)$, $x ∈ X$, and sequences $x ∈ X$ in $X$ and $y ∈ Y$ in $Y$ such that $x_{•} = (x_{i})∞ i=1$ in $X$ and $y_{•} = (y_{i})∞ i=1$ in $Y$, and $x_{•} → x$ for all $i$, then $y_{•} → y$.

Sufficient conditions for a closed graph

 * If $y_{i} ∈ F(x_{i})$ is a continuous function between topological spaces and if $Y$ is Hausdorff then $f$&thinsp; has a closed graph in $y ∈ F(x)$.
 * Note that if $f : X → Y$ is a function between Hausdorff topological spaces then it is possible for $f$&thinsp; to have a closed graph in $X ×&thinsp;Y$ 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 $f : X → Y$ is a function between topological spaces whose graph is closed in $X ×&thinsp;Y$ and if $Y$ is a compact space then $f : X → Y$ is continuous.

Continuous but not closed maps

 * Let $X$ denote the real numbers $X ×&thinsp;Y$ with the usual Euclidean topology and let $Y$ denote $f : X → Y$ with the indiscrete topology (where note that $Y$ is not Hausdorff and that every function valued in $Y$ is continuous). Let $ℝ$ be defined by $ℝ$ and $f : X → Y$ for all $f(0) = 1$. Then $f(x) = 0$ is continuous but its graph is not closed in $x ≠ 0$.
 * If $X$ is any space then the identity map $f : X → Y$ is continuous but its graph, which is the diagonal $X ×&thinsp;Y$, is closed in $Id : X → X$ if and only if $X$ is Hausdorff. In particular, if $X$ is not Hausdorff then $Gr Id := { (x, x) : x ∈ X&thinsp;}$ is continuous but not closed.
 * If $X × X$ is a continuous map whose graph is not closed then $Y$ is not a Hausdorff space.

Closed but not continuous maps

 * Let $X$ and $Y$ both denote the real numbers $Id : X → X$ with the usual Euclidean topology. Let $f : X → Y$ be defined by $ℝ$ and $f : X → Y$ for all $f(0) = 0$. Then $f(x) = 1⁄x$ has a closed graph (and a sequentially closed graph) in $x ≠ 0$ but it is not continuous (since it has a discontinuity at $f : X → Y$).
 * Let $X$ denote the real numbers $X ×&thinsp;Y = ℝ^{2}$ with the usual Euclidean topology, let $Y$ denote $x = 0$ with the discrete topology, and let $ℝ$ be the identity map (i.e. $ℝ$ for every $Id : X → Y$). Then $Id(x) := x$ is a linear map whose graph is closed in $x ∈ X$ but it is clearly not continuous (since singleton sets are open in $Y$ but not in $X$).
 * Let $Id : X → Y$ be a Hausdorff TVS and let $X ×&thinsp;Y$ be a vector topology on $X$ that is strictly finer than $(X, 𝜏)$. Then the identity map $𝜐$ a closed discontinuous linear operator.

Closed linear operators
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 $X$ and $Y$ are topological vector spaces (TVSs) then we call a linear map $𝜏$ a closed linear operator if its graph is closed in $Id : (X, 𝜏) → (X, 𝜐)$.

Closed graph theorem
The closed graph theorem states that any closed linear operator $f : D(f) ⊆ X → Y$ between two F-spaces (such as Banach spaces) is continuous, where recall that if $X$ and $Y$ are Banach spaces then $X ×&thinsp;Y$ being continuous is equivalent to $f$ being bounded.

Basic properties
The following properties are easily checked for a linear operator $f : X → Y$ between Banach spaces:


 * If $A$ is closed then $f : X → Y$ is closed where $λ$ is a scalar and $f : D(f) ⊆ X → Y$ is the identity function;
 * If $f$ is closed, then its kernel (or nullspace) is a closed vector subspace of $X$;
 * If $f$ is closed and injective then its inverse $A − λId_{D(f)}$ is also closed;
 * A linear operator $f$ admits a closure if and only if for every $Id_{D(f)}$ and every pair of sequences $f&thinsp;^{−1}$ and $x ∈ X$ in $x_{•} = (x_{i})∞ i=1$ both converging to $x$ in $X$, such that both $y_{•} = (y_{i})∞ i=1$ and $D(f)$ converge in $Y$, one has $f(x_{•}) = (f(x_{i}))∞ i=1$.

Example
Consider the derivative operator $f(y_{•}) = (f(y_{i}))∞ i=1$ where $lim_{i → ∞} fx_{i} = lim_{i → ∞} fy_{i}$ is the Banach space of all continuous functions on an interval $A = d⁄dx$. If one takes its domain $X = Y = C([a, b])$ to be $[a, b]$, then $f$ is a closed operator, which is not bounded. On the other hand if $D(f)$, then $f$ will no longer be closed, but it will be closable, with the closure being its extension defined on $C^{1}([a, b])$.