The Baire Category Theorem

Closed Graph Theorem

Let and be Banach spaces and a linear map of . Then is bounded if and only if the graph of is closed.

Helinger-Toeplitz Theorem

Let be an everywhere defined linear operator on a Hilbert space such that holds for all . Then is bounded.

Proof By the closed graph theorem, it suffices to show is closed in . Suppose is a sequence in that converges to . Then and . By the continuity of the inner product, we have Therefore by triangle inequality, we have This shows that , hence , which implies that is closed.