Definition and notation: The graph of a functionf : 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 2Y or 𝒫(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 F : X → 2Y with domainX that is valued in 2Y. 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 → 2Y a set-valued function 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 → 2Y is a set-valued function 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 set-valued function F : X → 2Y defined by F(x) := { f(x) } for every x ∈ X, where F is called the canonical set-valued function induced by (or associated with) f.
Note that in this case, Gr f = Gr F.
Open and closed graph
We give the more general definition of when a Y-valued function or set-valued function defined on a subsetS 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 spaceX (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 set-valued function on S (i.e. f : S → Y or f : S → 2Y). X × Y will always be endowed with the product topology.
Definition:[4] 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 set-valued function induced by g (i.e. G : S → 2Y 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.
Closable maps and closures
Definition: We say that the function (resp. set-valued function) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) 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).
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).
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.
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:
f has a closed graph (in X × Y);
(definition) the graph of f, Gr f, is a closed subset of X × Y;
for every x ∈ X and netx• = (xi)i ∈ I in X such that x• → x in X, if y ∈ Y is such that the net f(x•) := (f(xi))i ∈ I → y in Y then y = f(x);[4]
Compare this to the definition of continuity in terms of nets, which recall is the following: for every x ∈ X and net x• = (xi)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 both X and Y are metrizable spaces then we may add to this list:
for all x ∈ X, y ∈ Y, and sequences x• = (xi)∞ i=1 in X and y• = (yi)∞ i=1 in Y such that x• → x in X and y• → y in Y, and yi ∈ F(xi) for all i, then y ∈ F(x).[citation needed]
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.[4]
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.
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.[4]
Examples
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.[4]
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.[7] 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.
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).[4]
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).[4]
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) is a closed discontinuous linear operator.[8]
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 mapf : D(f) ⊆ X → Y a closed linear operator if its graph is closed in X × Y.
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.
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 − λIdD(f) is closed where λ is a scalar and IdD(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 inversef−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• = (xi)∞ i=1 and y• = (yi)∞ i=1 in D(f) both converging to x in X, such that both f(x•) = (f(xi))∞ i=1 and f(y•) = (f(yi))∞ i=1 converge in Y, one has limi → ∞fxi = limi → ∞fyi.
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.[9]
On the other hand if D(f) = 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]).
See also
Almost open linear map – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
^Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN0-471-50731-8.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN978-3-642-64988-2. MR0248498. OCLC840293704.