Equivalence of norms

Norms are equivalent iff they induce the same topology

Let be a vector space and be norms. Let and be the topologies on induced by and respectively. Then and are equivalent norms iff . topology

Proof

First suppose that and are equivalent, i.e. there exist such that for all . Now suppose is open under . Then for every there exists some such that . But , and thus for every there exists such that . Hence is open under . Therefore Since equivalence of norms is symmetric, by the same argument , and thus .

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