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lint by reynald, tentative near
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mkerjean committed Nov 16, 2020
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250 changes: 138 additions & 112 deletions theories/normedtype.v
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Expand Up @@ -4153,47 +4153,105 @@ rewrite !near_simpl near_withinE near_simpl => Pf; near=> y.
by have [->|] := eqVneq y x; [by apply: nbhs_singleton|near: y].
Grab Existential Variables. all: end_near. Qed.

(* NB: technical lemma that comes in handy in Banach-Steinhaus' PR *)
Lemma normrZV (R : numFieldType) (V : normedModType R) (x : V) :
x != 0 -> `| `| x|^-1 *: x | = 1.
Proof.
move=> x0; rewrite normmZ normrV ?unitfE ?normr_eq0 //=.
by rewrite normr_id mulVr // unitfE normr_eq0.
Qed.

Section LinearContinuousBounded.

Variables (R: numFieldType) (V W: normedModType R).
Variables (R : numFieldType) (V W : normedModType R).


(*drafts for another PR *)
Definition absorbed (A B : set V) :=
exists (r : R), A `<=` [ set x : V | exists y : V, B y /\ x = r *: y ].

Definition strong_bornology :=
[ set A | (forall (B : set V), (nbhs (0:V) B -> absorbed A B))].

Lemma normed_bornological (A : set V) : bounded_set A <->
(forall (B : set V), (nbhs (0:V) B -> absorbed A B)).
Proof.
Admitted.
(*To be generalized to any topological space, with a theory of bounded*)


Definition bounded_fun_on (f : V -> W) (B : set (set V)):=
forall A, (B A -> (exists M : R, \forall x \near (0 :V), (`|f x| <= M))).


Lemma linear_bounded (f : {linear V -> W} ):
( bounded_on f (nbhs (0 : V))) <->
(*exists r : R, 0 <= r /\ (forall x : V, `|f x| <= r * `|x|).*)
\forall r \near +oo, (forall x : V, `|f x| <= r * `|x|).
Proof.
split; last first.
move=> /pinfty_ex_gt0 [r [r0 Bf]]; rewrite /bounded_on; near=> M; rewrite -nearE.
near=> x; apply: le_trans; first by apply: Bf.
apply: (le_trans (y :=r)).
apply: ler_pimulr; apply: ltW; first by [].
near: x; rewrite nearE.
suff <-: ball_ normr 0 1 = (fun x : V => `|x| < 1) by apply: nbhs_ball_norm.
by apply: funext => x /=; rewrite sub0r normrN.
near: M; rewrite nearE /=; exists r; split; first by apply: gtr0_real.
by move => x; apply: ltW. (* much longer *)
rewrite /bounded_on => Bf; near=> r => x.
have [->|x0] := eqVneq x 0; first by rewrite linear0 !normr0 mulr0.
rewrite -ler_pdivr_mulr ; last by rewrite normr_gt0.
rewrite -(normr_id x) mulrC.
rewrite -normrV ?unitfE ?normr_eq0 // -normmZ -linearZ.
move:x x0; near: r.
(* hell on earth *)
(* how to use hypothesis with near *)
(* split => [[M /nbhs_ballP[a a0 Ba0]]|[r [r0 fr]]]; last first. *)
(* exists r; apply/nbhs_ballP; exists 1 => // x. *)
(* rewrite -ball_normE //= sub0r normrN => x1; apply: (le_trans (fr _)). *)
(* by rewrite ler_pimulr // ltW. *)
(* exists (M * 2 * a^-1); split. *)
(* - rewrite !mulr_ge0 //= ?invr_ge0 ; last by apply: ltW. *)
(* by move: (Ba0 0 (ballxx 0 a0)); rewrite linear0 normr0. *)
(* - move=> x; rewrite mulrAC ler_pdivl_mulr //=. *)
(* have [/eqP ->|x0] := boolP (x == 0). *)
(* + by rewrite linear0 !(normr0,mulr0,mul0r). *)
(* + rewrite mulrAC -lter_pdivr_mulr // -lter_pdivr_mulr ?normr_gt0 //. *)
(* rewrite [X in X <= _](_ : _ = `|f (a / 2 / `|x| *: x)|); last first. *)
(* rewrite linearZ normmZ 2!normrM (gtr0_norm a0) gtr0_norm //. *)
(* by rewrite normrV ?unitfE ?normr_eq0 // normr_id -2!mulrA mulrC mulrA. *)
(* apply: Ba0; rewrite -ball_normE /= sub0r normrN. *)
(* rewrite normmZ normrM (@normrV _ `|x|) ?unitfE ?normr_eq0 // normr_id. *)
(* rewrite -mulrA mulVr ?mulr1 ?unitfE ?normr_eq0 //. *)
(* by rewrite normrM 2?gtr0_norm// gtr_pmulr // invf_lt1 // ltr1n. *)
(* Qed. *)
Admitted.


Lemma linear_boundedP (f: {linear V -> W}) : bounded_on f (nbhs (0:V)) <->
(exists r, 0 <= r /\ (forall x : V, `|f x| <= `|x| * r )).
Lemma linear_boundedP (f : {linear V -> W}) :
(bounded_on f (nbhs (0 : V))) <->
exists r : R, 0 <= r /\ (forall x : V, `|f x| <= r * `|x|).
Proof.
split.
- move => /ex_bound [M /nbhs_ballP [a a0 Ba0]] //=.
exists (M * 2 *a^-1).
split.
- rewrite !mulr_ge0 //= ?invr_ge0 ; last by apply: ltW.
move : (Ba0 0 (ballxx 0 a0 )); rewrite linear0 normr0 //=.
- move => x. rewrite mulrA ler_pdivl_mulr //=.
case: (boolp.EM (x=0)).
- by move => ->; rewrite linear0 !normr0 !mul0r.
- move => /eqP x0 ; rewrite -lter_pdivr_mull ?normr_gt0 //=.
rewrite [X in ( _ <= X)]mulrC -lter_pdivr_mull //=.
have ->: 2^-1 *(`|x|^-1 * (`|f x| * a))=`|f ((2^-1 * `|x|^-1 * a)*:x)|.
rewrite linearZ normmZ !normrM !ger0_norm ?invr_ge0 //= ?ltW //=.
by rewrite mulrA mulrA mulrAC.
apply: Ba0 ;rewrite -ball_normE /= sub0r normrN ?normr_gt0 //=.
rewrite normmZ !normrM !normrV ?unitfE ?normrE // !ger0_norm ?ltW //=.
rewrite mulrAC -mulrA -mulrA [_ * (_ * a)]mulrA.
rewrite mulVr ?unitf_gt0 //= ?normr_gt0 //= mul1r.
rewrite mulrC gtr_pmulr ?invr_lte1 //= ?unitfE ?ltr_spaddr //=.
- move => [r [r0 fr]]; apply/ex_bound; exists r.
rewrite nbhs_ballP; exists 1; first by [].
move=> x; rewrite -ball_normE //= sub0r normrN => x1; apply: le_trans.
apply: fr.
rewrite -[X in _ <= X]mul1r; apply: ler_pmul; rewrite ?normr_ge0 //=.
by apply: ltW.
Qed.
split => [/linear_bounded | B].
by move=> /pinfty_ex_gt0 [M [M0 B]]; exists M; split; first by apply: ltW.
apply/linear_bounded; near_simpl. Admitted.

Lemma linear_continuous0 (f : {linear V -> W}) :
{for 0, continuous f} -> bounded_on f (nbhs (0:V)).
Proof.
move=> /cvg_ballP /(_ _ ltr01) .
{for 0, continuous f} -> bounded_on f (nbhs (0 : V)).
Proof.
(* move=> /cvg_distP; rewrite linear0 /= => Bf. *)
(* apply/linear_boundedP; near=> r => x. *)
(* have [->|x0] := eqVneq x 0; first by rewrite linear0 !normr0 mulr0. *)
(* rewrite -ler_pdivr_mulr; last by rewrite normr_gt0. *)
(* rewrite -(normr_id x) mulrC. *)
(* rewrite -normrV ?unitfE ?normr_eq0 // -normmZ -linearZ. *)
(* apply: ltW. move: (Bf r) x x0; near: r. apply/near_pinfty_div2. *)
move=> /cvg_ballP /(_ _ ltr01).
rewrite linear0 /= nearE => /nbhs_ex[tp ball_f]; apply/linear_boundedP.
pose t := tp%:num; exists (2 * t^-1); split => // x.
have [->|x0] := eqVneq x 0; first by rewrite linear0 !normr0 mul0r.
pose t := tp%:num; exists (2 * t^-1); split => [|x].
by apply: divr_ge0.
have [->|x0] := eqVneq x 0; first by rewrite linear0 !normr0 mulr0.
have /ball_f : ball 0 t ((`|x|^-1 * t /2) *: x).
apply: ball_sym; rewrite -ball_normE /ball_ subr0 normmZ mulrC 2!normrM.
rewrite 2!mulrA normrV ?unitfE ?normr_eq0 // normr_id.
Expand All @@ -4202,110 +4260,78 @@ have /ball_f : ball 0 t ((`|x|^-1 * t /2) *: x).
rewrite -ball_normE //= sub0r normrN linearZ /= normmZ -mulrA normrM.
rewrite normrV ?unitfE ?normr_eq0 // normr_id -mulrA.
rewrite ltr_pdivr_mull ?mulr1 ?normr_gt0 // -ltr_pdivl_mull ?normr_gt0 //.
by rewrite gtr0_norm // invf_div mulrC => /ltW.
rewrite gtr0_norm // invf_div => /ltW H. apply: le_trans; first by exact: H.
by apply: ler_pmul.
Qed.

Lemma linear_bounded0 (f : {linear V -> W}) :
bounded_on f (nbhs (0:V)) -> {for 0, continuous f} .
bounded_on f (nbhs (0 : V)) -> {for 0, continuous f} .
Proof.
move=> /linear_boundedP [r]; rewrite le0r=> [[/orP r0]]; case: r0.
- move/eqP => -> fr; apply: near_cst_continuous; near=> y.
by move: (fr y); rewrite mulr0 normr_le0; apply/eqP.
by move: (fr y); rewrite mul0r normr_le0; apply/eqP.
- move => r0 fr; apply/cvg_ballP => e e0.
rewrite nearE linear0 /= nbhs_ballP.
exists (e / 2 / r); first by rewrite !divr_gt0.
move=> x; rewrite -2!ball_normE /= 2!sub0r 2!normrN => xr.
have /le_lt_trans -> // : `|f x| <= e / 2.
by rewrite (le_trans (fr x)) // -ler_pdivl_mulr // ltW.
by rewrite (le_trans (fr x)) // mulrC -ler_pdivl_mulr // ltW.
by rewrite gtr_pmulr // invr_lt1 // ?unitfE // ltr1n.
Grab Existential Variables. by end_near.
Qed.
Grab Existential Variables. by end_near. Qed.

Lemma continuousfor0_continuous (f : {linear V -> W}) :
{for 0, continuous f} -> continuous f.
Proof.
move=> /(linear_continuous0) /linear_boundedP [r].
rewrite le0r=> [[/orP r0]]; case: r0 => /eqP.
- move => ->; move : f => [f linf] f0 //=. (* Linear linf = f ?? *)
suff: f = fun x => 0 by move => ->; apply: cst_continuous.
by apply: funext => x; move: (f0 x); rewrite mulr0 normr_le0; apply/eqP .
- move/eqP => //= r0 fr x.
rewrite cvg_to_locally => e e0; rewrite nearE /= nbhs_ballP.
exists (e / r).
- by rewrite mulr_gt0 //= invr_gt0.
- move=> y; rewrite -!ball_normE //= => xy; rewrite -linearB.
by rewrite (le_lt_trans (fr (x - y))) // -ltr_pdivl_mulr.
move=> /linear_continuous0 /linear_boundedP [r []].
rewrite le_eqVlt => /orP[/eqP <- f0|r0 fr x].
- suff : f = (fun _ => 0) :> (_ -> _) by move=> ->; exact: cst_continuous.
by rewrite funeqE => x; move: (f0 x); rewrite mul0r normr_le0 => /eqP.
- rewrite cvg_to_locally => e e0; rewrite nearE /= nbhs_ballP.
exists (e / r); first by rewrite divr_gt0.
move=> y; rewrite -!ball_normE //= => xy; rewrite -linearB.
by rewrite (le_lt_trans (fr (x - y))) // mulrC -ltr_pdivl_mulr.
Qed.

Lemma linear_bounded_continuous (f : {linear V -> W}) :
bounded_on f (nbhs (0 : V)) <-> continuous f.
Proof.
split; first by move=> /linear_bounded0; apply continuousfor0_continuous.
by move => /(_ 0) /linear_continuous0 /=.
split=> [/linear_bounded0|/(_ 0)/linear_continuous0//].
exact: continuousfor0_continuous.
Qed.

Definition bounded_fun_norm (f: V -> W) := forall r, exists M,
forall x, (`|x| <= r) -> (`|(f x)| <= M).
Definition bounded_fun_norm (f : V -> W) :=
forall r, exists M, forall x, `|x| <= r -> `|f x| <= M.
(*bounded_fun_on bounded_sets is harder to use*)

Definition pointwise_bounded (F: (V -> W) -> Prop) := forall x, exists M,
forall f , F f -> (`|f x| <= M)%O.
(* Definition pointwise_bounded (F : set (V -> W)) := *)
(* forall x, exists M, forall f, F f -> `|f x| <= M. *)

Definition uniform_bounded (F: (V -> W) -> Prop) := forall r, exists M,
forall f, F f -> forall x, (`|x| <= r) -> (`|f x| <= M)%O.
(* Definition uniform_bounded (F : set (V -> W)) := *)
(* forall r, exists M, forall f, F f -> forall x, `|x| <= r -> `|f x| <= M. *)

Lemma bounded_funP (f : {linear V -> W}):
bounded_fun_norm f <-> bounded_on f (nbhs (0:V)).
Proof.
split.
- move => /(_ 1) [M Bf]; apply /linear_boundedP.
have M0: (0 <= M) by move: (Bf 0); rewrite linear0 !normr0//= => /(_ ler01).
exists M; split; first by [].
move: M0; rewrite le0r => /orP; case.
- move=> /eqP M0; move: M0 Bf => -> Bf x; rewrite mulr0 normr_le0.
case: (EM (`|x| = 0)); move=>/eqP; rewrite normr_eq0=> /eqP x0.
- by rewrite x0 linear0.
- have x00: x != 0 by apply/eqP.
have : `| `|x|^-1 *: x | <= 1.
rewrite normmZ normrV ?unitf_gt0 ?normrE //=.
by rewrite mulrC mulrV ?unitf_gt0 ?normrE //=.
move=> /Bf; rewrite linearZ normr_le0 scaler_eq0 invr_eq0 => /orP.
by case ; rewrite ?normr_eq0 //=; move => /eqP ->; rewrite linear0.
- move => M0 x.
case: (EM ( x = 0)).
- by move=> ->;rewrite normr0 mul0r normr_le0 linear0.
- move => /eqP x0; rewrite mulrC -ler_pdivr_mulr ?normr_gt0 //= mulrC.
have <- : `| (`|x|^-1) | = `|x|^-1.
by apply/eqP; rewrite eqr_norm_id invr_ge0 normr_ge0.
rewrite -normmZ -linearZ; apply: (Bf (`|x|^-1 *: x)).
rewrite normmZ normrV ?unitf_gt0 ?normrE //=.
rewrite mulrC mulrV ?unitf_gt0 ?normrE //=.
- move => /linear_boundedP [r [r0 Bf]]; rewrite /bounded_fun_norm => e.
exists (e * r) => x xe; apply: le_trans; first by apply: Bf.
by apply: ler_pmul; rewrite ?normr_ge0 //=.
Qed.



(* Lemma bounded_landau (f :{linear V->W}) : *)
(* bounded_fun_norm f <-> ((f : V -> W) =O_ (0:V) cst (1 : K^o)). *)
(* Proof. *)
(* split. *)
(* - rewrite eqOP => bf. *)
(* move: (bf 1) => [M bm]. *)
(* rewrite !nearE /=; exists M; split. by apply : num_real. *)
(* move => x Mx; rewrite nearE nbhs_normP /=. *)
(* exists 1; first by []. *)
(* move => y /=. rewrite -ball_normE /ball_ sub0r normrN /cst normr1 mulr1 => y1. *)
(* apply: (@le_trans _ _ M _ _). *)
(* apply: (bm y); by apply: ltW. *)
(* by apply: ltW. *)
(* - rewrite eqOP /= ; move => Bf; apply/bounded_funP; rewrite /bounded_on. *)
(* near=>M. *)
(* set P := (X in (nbhs _ X)). *)
(* have -> : P = (fun x : V => (`|f x| <= M * `|cst 1%R x|)%O). *)
(* by apply: funext => x; rewrite /cst normr1 mulr1. *)
(* by near: M. *)
(* Grab Existential Variables. all: end_near. Qed. *)

bounded_fun_norm f <-> bounded_on f (nbhs (0 : V)).
Proof.
split => [/(_ 1) [M Bf]|/linear_boundedP [r [r0 Bf]] e]; last first.
by exists (e * r) => x xe; rewrite (le_trans (Bf _)) // mulrC ler_pmul.
apply/linear_boundedP.
have M0 : 0 <= M by move: (Bf 0); rewrite linear0 !normr0 => /(_ ler01).
exists M; split => //.
move: M0; rewrite le0r => /orP[/eqP|] M0 x.
- rewrite M0 mul0r normr_le0.
have [/eqP ->|x0] := boolP (x == 0); first by rewrite linear0.
have /Bf : `| `|x|^-1 *: x | <= 1.
rewrite normmZ normrV ?unitf_gt0 ?normrE //=.
by rewrite mulrC mulrV ?unitf_gt0 ?normrE.
rewrite linearZ M0 normr_le0 scaler_eq0 invr_eq0 => /orP[|//].
by rewrite normr_eq0 => /eqP ->; rewrite linear0.
- have [/eqP ->|x0] := boolP (x == 0).
+ by rewrite normr0 mulr0 normr_le0 linear0.
+ rewrite -ler_pdivr_mulr ?normr_gt0 //= mulrC.
have xx1 : `| `|x|^-1 | = `|x|^-1.
by rewrite normrV ?unitfE ?normr_eq0 // normr_id.
rewrite -xx1 -normmZ -linearZ; apply: (Bf (`|x|^-1 *: x)).
by rewrite normmZ xx1 mulVr // unitfE normr_eq0.
Qed.

End LinearContinuousBounded.

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