Closed graph theorem

From Academic Kids

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 { (x,y) ∈ X×Y | y = T(x) }.

Suppose that X and Y are Banach spaces, and that T is an everywhere-defined linear operator (i.e. the domain D(T) of T is X). Then T is continuous if and only if 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.

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. If the sequence {xn} in X converges to some element x, then the sequence {T(xn)} in Y also converges, and its limit is T(x).
  2. If the sequence {xn} in X converges to some element x and the {T(xn)} 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.

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