closed graph theorem
The closed graph theorem offers a labor-saving shortcut for proving an operator is continuous. Normally to show T is continuous you must check that x_n -> x forces T x_n -> T x — both the convergence of inputs and the convergence of outputs to the right place. This theorem lets you assume the outputs already converge to something, and only verify that the something is the correct value T x.
Precisely: let X and Y be Banach spaces and T : X -> Y a linear operator. Suppose its graph — the set of pairs (x, T x) — is closed in X (×) Y, meaning whenever x_n -> x and T x_n -> y, it follows that y = T x. Then T is bounded (continuous). In short, for linear maps between Banach spaces, a closed graph is enough; you get continuity for free.
The power lies in the weaker hypothesis. Proving continuity directly demands producing the limit of T x_n and identifying it; the closed-graph route grants you a limit y to work with and asks only to confirm y = T x, often far easier. As with the open mapping theorem, completeness of both spaces is essential; the result is in fact equivalent to the open mapping theorem and likewise rests on Baire category.
Suppose T : X -> Y is linear between Banach spaces and you know two facts: x_n -> 0, and T x_n -> y for some y. To apply the theorem you must show y = 0. If, say, you can independently prove T x_n -> 0 for every null sequence (perhaps via an integral identity), the graph is closed and T is bounded — no epsilon–delta estimate of the operator norm required.
Using the closed graph as a shortcut: assume the output limit exists, only identify it.
Continuity always implies a closed graph (in a Hausdorff target); the theorem supplies the nontrivial converse, valid only under completeness. The classic non-example is differentiation: as an operator on a dense subspace of L^2 it has a closed graph yet is unbounded — because it is defined on a non-closed domain, not on all of a Banach space.