Closed graph theorem explained

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.

A T. Tao’s blog post^[1] lists several closed graph theorems throughout mathematics.

Graphs and maps with closed graphs

See main article: Closed graph.

If

f : X \to Y is a map between topological spaces then the graph of f is the set \Gamma _{f} : = \{ (x, f(x)) : x \in X \} or equivalently,$$\backslash Gamma\_f\; :=\; \backslash$$It is said that the graph of f is closed if \Gamma _{f} is a closed subset of X x Y (with the product topology).

Any continuous function into a Hausdorff space has a closed graph (see)

Any linear map,

L : X \to Y, 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,, 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 x Y , then L is continuous and therefore necessarily sequentially continuous.

Examples of continuous maps that do not have a closed graph

If

X is any space then the identity map \operatorname{Id} : X \to X is continuous but its graph, which is the diagonal \Gamma _{\operatorname{Id} } := \,, is closed in X x X if and only if X is Hausdorff. In particular, if X is not Hausdorff then \operatorname{Id} : X \to X is continuous but does have a closed graph.

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 Hausdorff and that every function valued in Y is continuous). Let f : X \to Y be defined by f( 0 ) = 1 and f(x) = 0 for all x ≠ 0 . Then f : X \to Y is continuous but its graph is closed in X x Y .

Closed graph theorem in point-set topology

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

If X, Y are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see .

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact

Y is the real line, which allows the discontinuous function with closed graph f(x) = \begin{cases} (1)/(x) if x ≠ 0 ,\\ 0 else \end{cases} .

Also, closed linear operators in functional analysis (linear operators with closed graphs) are typically not continuous.

For set-valued functions

In functional analysis

See main article: Closed graph theorem (functional analysis).

If

T : X \to 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 x Y when X 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.

The theorem is a consequence of the open mapping theorem; see below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).

Relation to the open mapping theorem

Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way.^[2] Let

f : X \to Y be any map. Then it factors asf: X \overset{i}\to \Gamma _{f} \overset{q}\to Y .Now, i is the inverse of the projection p: \Gamma _{f} \to X . So, if the open mapping theorem holds for p ; i.e., p is an open mapping, then i is continuous and then f is continuous (as the composition of continuous maps).

For example, the above argument applies if

f is a linear operator between Banach spaces with closed graph, or if f is a map with closed graph between compact Hausdorff spaces.

Notes and References

1. https://terrytao.wordpress.com/2012/11/20/the-closed-graph-theorem-in-various-categories/
2. https://arxiv.org/abs/2403.03904