Closed graph theorem

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.

The closed graph theorem

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

$\lbrace (x,y) \in X\times Y \mid Tx=y\rbrace.$

If X is any topological space and Y is Hausdorff, then it is straightforward to show that the graph of T is closed whenever T is continuous.

If X and Y are Banach spaces, and T is an everywhere-defined (i.e. the domain D(T) of T is X) linear operator, then the converse is true as well. This is the content of the closed graph theorem: if the graph of T is closed in X × Y (with the product topology), we say that T is a closed operator, and, in this setting, we may conclude that T is continuous.

The restriction on the domain is needed due to the existence of closed unbounded linear operators. The differentiation operator on C([0,1]) is a prototypical counter-example.

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.

The closed graph theorem can be reformulated as follows. If T : X → Y is a linear bounded operator between Banach spaces, then the following are equivalent:

1. For every sequence {x_n} in X, if the sequence {x_n} converges in X to some element x, then the sequence {T(x_n)} in Y also converges, and its limit is T(x).
2. For every sequence {x_n} in X, if the sequence {x_n} converges in X to some element x and the sequence {T(x_n)} in Y converges to some element y, then y = T(x).

Generalization

The closed graph theorem can be generalized to more abstract topological vector spaces in the following way:

A linear operator from a barrelled space X to a Fréchet space Y is continuous if and only if its graph is closed in the space X×Y equipped with the product topology.

See also

• Discontinuous linear map

References

• Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1st ed.), John Wiley & Sons, ISBN 978-0-471-80958-6 
• Rudin, Walter (1973), Functional analysis, Tata MacGraw-Hill .
• Proof of closed graph theorem on PlanetMath