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).
Definition prod_norm_factor :=
sqrt 2 * Rmax (@norm_factor K U) (@norm_factor K V).
End prod_NormedModule.
Definition prod_NormedModule_mixin (K : AbsRing) (U V : NormedModule K) :=
NormedModule.Mixin K _ (@prod_norm K U V) prod_norm_factor prod_norm_triangle
prod_norm_scal prod_norm_compat1 prod_norm_compat2 prod_norm_eq_zero.
Canonical prod_NormedModule (K : AbsRing) (U V : NormedModule K) :=
NormedModule.Pack K (U * V) (NormedModule.Class K (U * V) _ (prod_NormedModule_mixin K U V)) (U * V).

Lemma norm_prod {K : AbsRing}
{U : NormedModule K} {V : NormedModule K}
(x : U) (y : V) :
Rmax (norm x) (norm y) <= norm (x,y) <= sqrt 2 * Rmax (norm x) (norm y).