, 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