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