Section LinearFct.
(** * Linear functions *)
Context {K : AbsRing} {U V : NormedModule K}.
Record is_linear (l : U -> V) := {
linear_plus : forall (x y : U), l (plus x y) = plus (l x) (l y) ;
linear_scal : forall (k : K) (x : U), l (scal k x) = scal k (l x) ;
linear_norm : exists M : R, 0 < M /\ (forall x : U, norm (l x) <= M * norm x) }.

(** zero in a linear function *)
Lemma is_linear_zero : is_linear (fun _ => zero).