The Closed graph theorem reference article from the English Wikipedia on 24-Jul-2004
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Closed graph theorem

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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 {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).