Closed graph theorem

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In particular, they give conditions when functions with closed graphs are necessarily continuous. In mathematics, there are several results known as the "closed graph theorem".

This article is about closed graph theorems in general topology. For the closed graph theorem in functional analysis, see Closed graph theorem (functional analysis).
The graph of the cubic function f(x) = x3 − 9x on the interval [-4,4] is closed because the function is continuous. The graph of the Heaviside function on [-2,2] is not closed, because the function is not continuous.

Graphs and maps with closed graphs
Main article: Closed graph

If f : X → Y is a map between topological spaces then the graph of f is the set Gr f := { (x, f(x)) : x ∈ X } or equivalently,

Gr f := { (x, y) ∈ X × Y : y = f(x) }

We say that the graph of f is closed if Gr f is a closed subset of X × Y (with the product topology).

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

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

Examples of continuous maps that are not closed
  • 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.[2] In particular, if X is not Hausdorff then Id : X → X is continuous but not closed.
  • 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.[3]

Closed graph theorem in point-set topology

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

Closed graph theorem[4] — If f : X → Y is a map from a topological space X into a compact Hausdorff space Y, then the graph of f is closed if and only if f : X → Y is continuous.

For set-valued functions

Closed graph theorem for set-valued functions[5] — For a Hausdorff compact range space Y, a set-valued function F : X → 2Y has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all x ∈ X.

In functional analysis
Main article: Closed graph theorem (functional analysis)
Definition: If T : X → Y is a linear operator between topological vector spaces (TVSs) then we say that T is a closed operator if the graph of T is closed in X × Y when X × Y 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

Notes
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
  • Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1st ed.), John Wiley & Sons, ISBN 978-0-471-80958-6
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
  • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
  • Munkres, James R. (January 7, 2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
  • "Proof of closed graph theorem". PlanetMath.

References
  1. Rudin 1991, p. 51-52.
  2. Rudin 1991, p. 50.
  3. Narici 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 1999, p. 78.
  7. Trèves (1995), p. 173
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