Closed graph theorem - Definition and Overview

In mathematics, the closed graph theorem is a basic result of functional analysis.

For any function T : X → Y, we define the graph of T to be the set { (x,y) ∈ X×Y | y = T(x) }.

The closed graph theorem states the following. Suppose that X and Y are Banach spaces, and that T is a linear operator. Then T is continuous if and only if its graph is closed in X×Y (with the product topology).

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