Section Op_LinearFct.
Context {K : AbsRing} {V : NormedModule K}.

(* Specifications: *)
(** id is a linear function *)
Lemma is_linear_id : is_linear (fun (x : V) => x).
(** opp is a linear function *)
Lemma is_linear_opp : is_linear (@opp V).
(** plus is a linear function *)
Lemma is_linear_plus : is_linear (fun x : V * V => plus (fst x) (snd x)).
(** [fun k => scal k x] is a linear function *)
Lemma is_linear_scal_l (x : V) :
is_linear (fun k : K => scal k x).
(** [fun x => scal k x] is a linear function if [mult] is commutative *)
Lemma is_linear_scal_r (k : K) :
(forall n m : K, mult n m = mult m n)
-> is_linear (fun x : V => scal k x).