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 . #m/thm/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 .