In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.[1][2] A function f : XY between topological spaces has a closed graph if its graph is a closed subset of the product space X ×Y. 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 f : XY is the set
Gr f := { (x, f(x)) : xX} = { (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 domain X that is valued in 2Y. That is, F is a function on X such that for every xX, 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 xX; 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 : yF(x)}.
Definition: A function f : XY can be canonically identified with the set-valued function F : X → 2Y defined by F(x) := { f(x)} for every xX, 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 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, SX, and f is a Y-valued function or set-valued function on S (i.e. f : SY 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 : SY 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 sS) 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 DX containing S and a function (resp. set-valued function) F : DY 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 : SY 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 DS such that the closure in X ×Y of the graph of the restriction f|D : DY 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) ⊆ XY 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) ⊆ XY 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 : XY 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 : XY 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 : XY 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 : XY 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 xX and net x = (xi)iI in X such that xx in X, if yY is such that the net f(x) := (f(xi))iIy 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 xX and net x = (xi)iI in X such that xx 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 yY (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 yY 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. f is continuous;[5]

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

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

If f : XY 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 xX and sequence x = (xi)
    i=1
    in X such that xx in X, if yY is such that the net f(x) := (f(xi))
    i=1
    y
    in Y then y = f(x);[4]
set-valued function with a closed graph

If F : X → 2Y 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. F is upper hemicontinuous and F(x) is a closed subset of Y for all xX;[6]

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

  1. for all xX, yY, and sequences x = (xi)
    i=1
    in X and y = (yi)
    i=1
    in Y such that xx in X and yy in Y, and yiF(xi) for all i, then yF(x).

Sufficient conditions for a closed graph

  • If f : XY 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 : XY 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 : XY is a function between topological spaces whose graph is closed in X ×Y and if Y is a compact space then f : XY is continuous.[4]

Examples

Continuous but not closed maps

  • Let X denote the real numbers with the usual Euclidean topology and let Y denote with the indiscrete topology (where note that Y is not Hausdorff and that every function valued in Y is continuous). Let f : XY be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : XY is continuous but its graph is not closed in X ×Y.[4]
  • If X is any space then the identity map Id : XX is continuous but its graph, which is the diagonal Gr Id := { (x, x) : xX}, is closed in X × X if and only if X is Hausdorff.[7] In particular, if X is not Hausdorff then Id : XX is continuous but not closed.
  • If f : XY 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 with the usual Euclidean topology. Let f : XY be defined by f(0) = 0 and f(x) = 1/x for all x ≠ 0. Then f : XY 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 with the usual Euclidean topology, let Y denote with the discrete topology, and let Id : XY be the identity map (i.e. Id(x) := x for every xX). Then Id : XY 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) a closed discontinuous linear operator.[8]

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) ⊆ XY 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 : XY between two F-spaces (such as Banach spaces) is continuous, where recall that if X and Y are Banach spaces then f : XY being continuous is equivalent to f being bounded.

Basic properties

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

  • If A is closed then AλIdD(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 inverse f−1 is also closed;
  • A linear operator f admits a closure if and only if for every xX 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 {{math|1=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]).

See also

References

  1. Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.
  2. Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.
  3. Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.
  4. 1 2 3 4 5 6 7 8 Narici & Beckenstein 2011, pp. 459–483.
  5. Munkres 2000, p. 171.
  6. Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
  7. Rudin p.50
  8. Narici & Beckenstein 2011, p. 480.
  9. Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8.
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