(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
Open Scope R_scope.
(** This file describes results about differentiability on a generic normed module. Specific results are also given on [R] with a total function [Derive]. It ends with the Taylor-Lagrange formula. *)
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) }.

Lemma linear_opp (l : U -> V) (x : U) : is_linear l ->
l (opp x) = opp (l x).