Closed graph theorem

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. There are several versions of the theorem.

In mathematics, there are several results known as the "closed graph theorem".

For any function T : X → Y, we define the graph of T to be the set

In point-set topology, the closed graph theorem states the following: If X is a topological space and Y is a compact Hausdorff space, then the graph of T is closed if and only if T is continuous.

The rest of the section concerns functional analysis, where the closed graph theorem states the following: If X and Y are Banach spaces, and T : X → Y is a linear operator, then T is continuous if and only if its graph is closed in X × Y (with the product topology).

In the latter case we say that T is a closed operator. Note that the operator is required to be everywhere-defined, i.e., the domain D(T) of T is X. This condition is necessary, as there exist closed linear operators that are unbounded (not continuous); a prototypical example is provided by the derivative operator on C([0,1]) (whose domain is a strict subset of C([0,1])).

The usual proof of the closed graph theorem employs the open mapping theorem. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the necessity of X and Y being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.

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