Closed graph theorem

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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.

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".

Contents

Graphs and maps with closed graphs

If f : XY is a map between topological spaces then the graph of f is the set Gr f := { (x, f(x)) : xX} 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 : XY, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (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

Closed graph theorem in point-set topology

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

Closed graph theorem [4]   If f : XY 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 : XY 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 xX.

In functional analysis

Definition: If T : XY 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

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In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals to linear operators valued in topological vector spaces (TVSs).

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map.

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 (1995) , p. 173

Notes

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