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) }.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) }.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) }.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) }.

Lemma linear_zero (l : U -> V) : is_linear l -> l zero = zero.