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 is_linear_ext (l1 l2 : U -> V) : (forall x, l1 x = l2 x) -> is_linear l1 -> is_linear l2.