The Baire Category Theorem

Baire

A Baire Space is a topological space in which every countable intersection of open dense subsets remains dense. In other words, is not a countable union of nowhere dense sets.

Baire Category Theorem

Every completely metrizable space is Baire. Every locally compact Hausdorff space is Baire.

Open Mapping Theorem

Open Mapping Theorem

Let and be Banach spaces and a surjective bounded linear map of . Then is an open map.

Corollary

Suppose is a bijective bounded linear map between Banach spaces. Then is bounded.

Proof Since is bijective, it is surjective. By the open mapping theorem, is an open map. Therefore, for any open set , we have is open in . This shows that is continuous, thus bounded.

Closed Graph 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 in .

Proof It is clear that if

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.

Uniform Boundedness Principle

Uniform Boundedness Principle

Let be a Banach space and a normed vector space. Let be a family of bounded linear operators from to . If for every , the set is bounded, then the set is bounded.

Proof Define sets . Each is closed because if a sequence converges to , then by continuity, so for all , thus for all , therefore, . Note that from our assumption, we have . By the Baire category theorem, there exists some such that has nonempty interior. So there exists and such that the closure of is contained in . For any arbitrary with , we have which proves the desired statement.