Comparability Theorem

Let and be complete norms on a vector space . Suppose for all and some . Then and are equivalent ( for all and some )

Proof

Let be the identity map from to Then continuous Also is a bijection So also continuous by Inversion Theorem.

Remarks

1. Comparability theorem gives a silly reason why is incomplete in
2. Inversion Theorem tells us that if , Banach Space, surjective then is isomorphic to .