Definition AbsRing_NormedModule_mixin :=
NormedModule.Mixin K _ abs 1 abs_triangle abs_mult (fun x y e H => H) AbsRing_norm_compat2 abs_eq_zero.
Canonical AbsRing_NormedModule :=
NormedModule.Pack K _ (NormedModule.Class _ _ _ AbsRing_NormedModule_mixin) K.
End AbsRing_NormedModule.
(** Normed vector spaces have some continuous functions *)
Section NVS_continuity.
Context {K : AbsRing} {V : NormedModule K}.

(* Specifications: *)
Lemma filterlim_plus :
forall x y : V,
filterlim (fun z : V * V => plus (fst z) (snd z)) (filter_prod (locally x) (locally y)) (locally (plus x y)).
Lemma filterlim_scal (k : K) (x : V) :
filterlim (fun z => scal (fst z) (snd z)) (filter_prod (locally k) (locally x)) (locally (scal k x)).
Lemma filterlim_scal_r (k : K) (x : V) :
filterlim (fun z : V => scal k z) (locally x) (locally (scal k x)).
Lemma filterlim_scal_l (k : K) (x : V) :
filterlim (fun z => scal z x) (locally k) (locally (scal k x)).
Lemma filterlim_opp :
forall x : V,
filterlim opp (locally x) (locally (opp x)).