The Baire Category Theorem

MORAL: Banach Spaces are very nice

Theorem

Let be a (nonempty) Banach Space space, and a sequence of dense open sets. Then:

Notation

For open balls write For closed balls write

Proof

so there is some some and some is dense so it meets so there is some for some and some Continue inductively and obtain sequence with and for all Now is a Cauchy Sequence and for any have Let Then for all .

Theorem (alt)

complete metric, closed subsets of with . Then some has

Theorem (altalt)

complete metric space with nowhere dense subsets of . Then .

Theorem (altaltalt)

If is complete, then is not a Meagre subset of .

Osgood’s Theorem Principle of Uniform Boundedness Banach-Stewhaus Theorem Open Mapping Theorem Continuous, Nowhere Differentiable Functions