1. Definitions

1.1. Graphs and Multifunctions

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

Gr f := { (x, f(x)) : x ∈ X } = { (x, y) ∈ X × Y : y = f(x) }.

Notation: If Y is a set then the power set of Y, which is the set of all subsets of Y, is denoted by 2^Y or 𝒫(Y).

Definition: If X and Y are sets then a Y-valued multifunction on X, also called a set-valued function in Y on X, is a function F : X → 2^Y with domain X that is valued in 2^Y. That is, F is a function on X such that for every x ∈ X, F(x) is a subset of Y.

• Some authors call a function F : X → 2^Y a multifunction only if it satisfies the additional requirement that F(x) is not empty for every x ∈ X; this article does not require this.

Definition and notation: If F : X → 2^Y is a multifunction in a set Y then the graph of F is the set

Gr F := { (x, y) ∈ X × Y : y ∈ F(x) }.

Definition: A function f : X → Y can be canonically identified with the multifunction F : X → 2^Y defined by F(x) := { f(x) } for every x ∈ X, where F is called the canonical multifunction induced by (or associated with) f.

• Note that in this case, Gr f = Gr F.

1.2. Open and Closed Graph

We give the more general definition of when a Y-valued function or multifunction 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, S ⊆ X, and f is a Y-valued function or multifunction on S (i.e. f : S → Y or f : S → 2^Y). X × Y will always be endowed with the product topology.

Definition:^[1] We say that f  has a closed graph (resp. open graph, sequentially closed graph, sequentially open graph) in X × Y if the graph of f, Gr f, is a closed (resp. open, sequentially closed, sequentially open) subset of X × Y when X × Y is endowed with the product topology. If S = X or if X is clear from context then we may omit writing "in X × Y"

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

1.3. Closable Maps and Closures

Definition: We say that the function (resp. multifunction) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. multifunction) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f.

• Additional assumptions for linear maps: If in addition, S, X, and Y are topological vector spaces and f : S → 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 D ⊆ S such that the closure in X × Y of the graph of the restriction f |_D : D → Y of f to D is equal to the closure of the graph of f in X × Y (i.e. the closure of Gr f in X × Y is equal to the closure of Gr f |_D in X × Y).

1.3. Closed Maps and Closed Linear Operators

Definition and notation: When we write f : D(f) ⊆ X → Y then we mean that f is a Y-valued function with domain D(f) where D(f) ⊆ X. If we say that f : D(f) ⊆ X → Y 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 X × Y (rather than in D(f) × Y).

When reading literature in functional analysis, if f : X → 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 f : X → Y is called closed if its graph is closed in 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.

2. Characterizations

Throughout, let X and Y be topological spaces.

Function with a closed graph

If f : X → Y is a function then the following are equivalent:

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

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

1.  
4. f  is continuous;^[2]

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

1.  
5. f  has a sequentially closed graph (in X × Y);

Function with a sequentially closed graph

If f : X → Y is a function then the following are equivalent:

1. f  has a sequentially closed graph (in X × Y);
2. (definition) the graph of f is a sequentially closed subset of X × Y;
3. for every x ∈ X and sequence x_• = (x_i)∞i=1 in X such that x_• → x in X, if y ∈ Y is such that the net f(x_•) := (f(x_i))∞i=1 → y in Y then y = f(x);^[1]

Multifunction with a closed graph

If F : X → 2^Y is a set-valued function between topological spaces X and Y then the following are equivalent:

1. F  has a closed graph (in X × Y);
2. (definition) the graph of F is a closed subset of X × Y;

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

1.  
3. F is upper hemicontinuous and F(x) is a closed subset of Y for all x ∈ X;^[3]

and if both X and Y are metrizable spaces then we may add to this list:

1.  
4. for all x ∈ X, y ∈ Y, and sequences x_• = (x_i)∞i=1 in X and y_• = (y_i)∞i=1 in Y such that x_• → x in X and y_• → y in Y, and y_i ∈ F(x_i) for all i, then y ∈ F(x).

3. Sufficient Conditions for a Closed Graph

• If f : X → Y is a continuous function between topological spaces and if Y is Hausdorff then f  has a closed graph in X × Y.^[1]
  • Note that if f : X → Y is a function between Hausdorff topological spaces then it is possible for f  to have a closed graph in X × Y but not be continuous.

4. 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 × Y and if Y is a compact space then f : X → Y is continuous.^[1]

5. Examples

5.1. Continuous but not Closed Maps

• Let X denote the real numbers ℝ with the usual Euclidean topology and let Y denote ℝ with the indiscrete topology (where note that Y is not Hausdorff and that every function valued in Y is continuous). Let f : X → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y.^[1]
• If X is any space then the identity map Id : X → X is continuous but its graph, which is the diagonal Gr Id := { (x, x) : x ∈ X }, is closed in X × X if and only if X is Hausdorff.^[4] In particular, if X is not Hausdorff then Id : X → X is continuous but not closed.
• If f : X → Y is a continuous map whose graph is not closed then Y is not a Hausdorff space.

5.2. Closed but not Continuous Maps

• Let X and Y both denote the real numbers ℝ with the usual Euclidean topology. Let f : X → Y be defined by f(0) = 0 and f(x) = 1/x for all x ≠ 0. Then f : X → Y has a closed graph (and a sequentially closed graph) in X × Y = ℝ² but it is not continuous (since it has a discontinuity at x = 0).^[1]
• Let X denote the real numbers ℝ with the usual Euclidean topology, let Y denote ℝ with the discrete topology, and let Id : X → Y be the identity map (i.e. Id(x) := x for every x ∈ X). Then Id : X → Y is a linear map whose graph is closed in X × Y but it is clearly not continuous (since singleton sets are open in Y but not in X).^[1]
• Let (X, 𝜏) be a Hausdorff TVS and let 𝜐 be a vector topology on X that is strictly finer than 𝜏. Then the identity map Id : (X, 𝜏) → (X, 𝜐) a closed discontinuous linear operator.^[5]

6. 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 f : D(f) ⊆ X → Y a closed linear operator if its graph is closed in X × Y.

6.1. Closed Graph Theorem

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

6.2. Basic Properties

The following properties are easily checked for a linear operator f : D(f) ⊆ X → Y between Banach spaces:

• If A is closed then A − λId_D(f) is closed where λ is a scalar and Id_D(f) 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 f ⁻¹ is also closed;
• A linear operator f admits a closure if and only if for every x ∈ X and every pair of sequences x_• = (x_i)∞i=1 and y_• = (y_i)∞i=1 in D(f) both converging to x in X, such that both f(x_•) = (f(x_i))∞i=1 and f(y_•) = (f(y_i))∞i=1 converge in Y, one has lim_{i → ∞} fx_i = lim_{i → ∞} fy_i.

6.3. Example

Consider the derivative operator A = d/dx where X = Y = C([a, b]) is the Banach space of all continuous functions on an interval [a, b]. If one takes its domain D(f) to be C¹([a, b]), then f is a closed operator, which is not bounded.^[6] On the other hand if D(f) = smooth function|C^∞([a, b]), then f will no longer be closed, but it will be closable, with the closure being its extension defined on C¹([a, b]).

The content is sourced from: https://handwiki.org/wiki/Closed_graph