Section prod_NormedModule.
Context {K : AbsRing} {U V : NormedModule K}.
Definition prod_norm (x : U * V) :=
sqrt (norm (fst x) ^ 2 + norm (snd x) ^ 2).

Lemma prod_norm_compat1 :
forall (x y : U * V) (eps : R),
prod_norm (minus y x) < eps -> ball x eps y.