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).

(* Specifications: *)
Lemma prod_norm_triangle :
forall x y : U * V,
prod_norm (plus x y) <= prod_norm x + prod_norm y.
Lemma prod_norm_scal :
forall (l : K) (x : U * V),
prod_norm (scal l x) <= abs l * prod_norm x.
Lemma prod_norm_compat1 :
forall (x y : U * V) (eps : R),
prod_norm (minus y x) < eps -> ball x eps y.