Closed graph theorem

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Short description

: Theorem relating continuity to graphs

A cubic function

The Heaviside function

The graph of the cubic function [math]\displaystyle{ f(x) = x^3 - 9x }[/math] on the interval [math]\displaystyle{ [-4, 4] }[/math] is closed because the function is continuous. The graph of the Heaviside function on [math]\displaystyle{ [-2, 2] }[/math] is not closed, because the function is not continuous.

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.

Graphs and maps with closed graphs

Main page: Closed graph

If [math]\displaystyle{ f : X \to Y }[/math] is a map between topological spaces then the graph of [math]\displaystyle{ f }[/math] is the set [math]\displaystyle{ \operatorname{Gr} f := \{ (x, f(x)) : x \in X \} }[/math] or equivalently, [math]\displaystyle{ \operatorname{Gr} f := \{ (x, y) \in X \times Y : y = f(x) \} }[/math] It is said that the graph of [math]\displaystyle{ f }[/math] is closed if [math]\displaystyle{ \operatorname{Gr} f }[/math] is a closed subset of [math]\displaystyle{ X \times Y }[/math] (with the product topology).

Any continuous function into a Hausdorff space has a closed graph.

Any linear map, [math]\displaystyle{ L : X \to Y, }[/math] between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) [math]\displaystyle{ L }[/math] is sequentially continuous in the sense of the product topology, then the map [math]\displaystyle{ L }[/math] is continuous and its graph, Gr L , is necessarily closed. Conversely, if [math]\displaystyle{ L }[/math] is such a linear map with, in place of (1a), the graph of [math]\displaystyle{ L }[/math] is (1b) known to be closed in the Cartesian product space [math]\displaystyle{ X \times Y }[/math], then [math]\displaystyle{ L }[/math] is continuous and therefore necessarily sequentially continuous.[1]

Examples of continuous maps that do not have a closed graph

If [math]\displaystyle{ X }[/math] is any space then the identity map [math]\displaystyle{ \operatorname{Id} : X \to X }[/math] is continuous but its graph, which is the diagonal [math]\displaystyle{ \operatorname{Gr} \operatorname{Id} := \{ (x, x) : x \in X \}, }[/math], is closed in [math]\displaystyle{ X \times X }[/math] if and only if [math]\displaystyle{ X }[/math] is Hausdorff.[2] In particular, if [math]\displaystyle{ X }[/math] is not Hausdorff then [math]\displaystyle{ \operatorname{Id} : X \to X }[/math] is continuous but does not have a closed graph.

Let [math]\displaystyle{ X }[/math] denote the real numbers [math]\displaystyle{ \R }[/math] with the usual Euclidean topology and let [math]\displaystyle{ Y }[/math] denote [math]\displaystyle{ \R }[/math] with the indiscrete topology (where note that [math]\displaystyle{ Y }[/math] is not Hausdorff and that every function valued in [math]\displaystyle{ Y }[/math] is continuous). Let [math]\displaystyle{ f : X \to Y }[/math] be defined by [math]\displaystyle{ f(0) = 1 }[/math] and [math]\displaystyle{ f(x) = 0 }[/math] for all [math]\displaystyle{ x \neq 0 }[/math]. Then [math]\displaystyle{ f : X \to Y }[/math] is continuous but its graph is not closed in [math]\displaystyle{ X \times Y }[/math].[3]

Closed graph theorem in point-set topology

In point-set topology, the closed graph theorem states the following:

Closed graph theorem[4]  —If [math]\displaystyle{ f : X \to Y }[/math] is a map from a topological space [math]\displaystyle{ X }[/math] into a Hausdorff space [math]\displaystyle{ Y, }[/math] then the graph of [math]\displaystyle{ f }[/math] is closed if [math]\displaystyle{ f : X \to Y }[/math] is continuous. The converse is true when [math]\displaystyle{ Y }[/math] is compact. (Note that compactness and Hausdorffness do not imply each other.)

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact [math]\displaystyle{ Y }[/math] is the real line, which allows the discontinuous function with closed graph [math]\displaystyle{ f(x) = \begin{cases} \frac 1 x \text{ if }x\neq 0,\\ 0\text{ else} \end{cases} }[/math].

For set-valued functions

Closed graph theorem for set-valued functions[5]  —For a Hausdorff compact range space [math]\displaystyle{ Y }[/math], a set-valued function [math]\displaystyle{ F : X \to 2^Y }[/math] has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all [math]\displaystyle{ x \in X }[/math].

In functional analysis

Main page: Closed graph theorem (functional analysis)

If [math]\displaystyle{ T : X \to Y }[/math] is a linear operator between topological vector spaces (TVSs) then we say that [math]\displaystyle{ T }[/math] is a closed operator if the graph of [math]\displaystyle{ T }[/math] is closed in [math]\displaystyle{ X \times Y }[/math] when [math]\displaystyle{ X \times Y }[/math] is endowed with the product topology.

The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.

Theorem[6] [7]  —A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.

See also

  • Almost open linear map
  • Barrelled space – Topological vector space
  • Closed graph – Graph of a map closed in the product space
  • Discontinuous linear map
  • Kakutani fixed-point theorem – On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point
  • Open mapping theorem (functional analysis) – Condition for a linear operator to be open
  • Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
  • Webbed space – Spaces where open mapping and closed graphs theorems hold
  • Zariski's main theorem – Theorem of algebraic geometry and commutative algebra

Notes

References

  1. ↑ Rudin 1991, p. 51-52.
  2. ↑ Rudin 1991, p. 50.
  3. ↑ Narici & Beckenstein 2011, pp. 459-483.
  4. ↑ Munkres 2000, pp. 163–172.
  5. ↑ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
  6. ↑ Schaefer & Wolff 1999, p. 78.
  7. ↑ (Trèves 2006), p. 173

Bibliography

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