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 .