Open Mapping Theorem

, Banach Space, surjective. Then is an open mapping.
(ie st )

Proof

surjectivee
So with each closed.
Thus by The Baire Category Theorem, some contains a ball .
WLOG : so dense in .
For any and any :
Have

So there are some with and

Thus and
So dense in .
Thus any open mapping by Open Mapping Lemma applied to .

Corollaries

Inversion Theorem
Comparability Theorem