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

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 T is continuous if and only if it is a closed operator, that is, its graph is closed in X × Y (with the product topology).

The restriction on the domain is needed due to the existence of closed unbounded linear operators.

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.

The closed graph theorem can be reformulated as follows. If T : X → Y is a linear 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

• Proof of closed graph theorem on PlanetMath