From 3e76ee7daeb0bec0a37c68df1d3364299ea8aa4f Mon Sep 17 00:00:00 2001 From: Quentin Vermande Date: Sun, 24 Aug 2025 11:39:14 +0200 Subject: [PATCH 1/2] normed vector types, infinite norm, norm equivalence thm, continuity of linear functions in finite dimension --- CHANGELOG_UNRELEASED.md | 10 + theories/normedtype_theory/normed_module.v | 245 +++++++++++++++++++++ theories/tvs.v | 16 ++ 3 files changed, 271 insertions(+) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index a5849a8539..57b19d9108 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -32,6 +32,16 @@ + number notations in scopes `ereal_dual_scope` and `ereal_scope` + notation `- 1` in scopes `ereal_dual_scope` and `ereal_scope` +- in `tvs.v` + + lemmas `cvg_sum`, `sum_continuous` + +- in `normed_module.v`: + + structure `NormedVector` + + definitions `normedVectType`, `oo_norm`, `oo_space` + + lemmas `oo_norm_ge0`, `le_coord_oo_norm`, `ler_oo_normD`, `oo_norm0_eq0`, + `oo_normZ`, `oo_normMn`, `oo_normN`, `oo_closed_ball_compact`, + `equivalence_norms`, `linear_findim_continuous` + ### Changed - in `lebesgue_stieltjes_measure.v` specialized from `numFieldType` to `realFieldType`: diff --git a/theories/normedtype_theory/normed_module.v b/theories/normedtype_theory/normed_module.v index 6af95b7d03..1ac0300a38 100644 --- a/theories/normedtype_theory/normed_module.v +++ b/theories/normedtype_theory/normed_module.v @@ -20,6 +20,9 @@ From mathcomp Require Import ereal_normedtype pseudometric_normed_Zmodule. (* normedModType K == interface type for a normed module *) (* structure over the numDomainType K *) (* The HB class is NormedModule. *) +(* normedVectType K == interface type for a normed vectType *) +(* structure over the numDomainType K *) +(* The HB class is NormedVector. *) (* `|x| == the norm of x (notation from ssrnum.v) *) (* ``` *) (* *) @@ -147,6 +150,10 @@ HB.instance Definition _ := HB.end. +#[short(type="normedVectType")] +HB.structure Definition NormedVector (K : numDomainType) := + {T of NormedModule K T & Vector K T}. + (**md see also `Section standard_topology_pseudoMetricNormedZmod` in `pseudometric_normed_Zmodule.v` *) Section standard_topology_normedMod. @@ -2079,3 +2086,241 @@ by rewrite interior_closed_ballE //; exact: ballxx. Qed. End Closed_Ball_normedModType. + +Section InfiniteNorm. +Variables (R : numFieldType) (V : vectType R) (B : (\dim (@fullv _ V)).-tuple V). +Let V' := @fullv _ V. +Hypothesis (Bbasis : basis_of V' B). + +Definition oo_norm x := \big[Order.max/0]_(i < \dim V') `|coord B i x|. + +Definition oo_space : Type := (fun=> V) Bbasis. + +HB.instance Definition _ := Vector.on oo_space. + +HB.instance Definition _ := Pointed.copy oo_space V^o. + +Lemma oo_norm_ge0 x : 0 <= oo_norm x. +Proof. +rewrite /oo_norm. +elim: (index_enum _) => /=; first by rewrite big_nil. +by move=> i l IHl; rewrite big_cons /Order.max/=; case: ifP. +Qed. + +Lemma le_coord_oo_norm x i : `|coord B i x| <= oo_norm x. +Proof. +rewrite /oo_norm; elim: (index_enum _) (mem_index_enum i) => //= j l IHl. +rewrite inE big_cons /Order.max/= => /orP[/eqP <-|/IHl {}IHl]; + case: ifP => [/ltW|]//. +move=> /negP/negP. +have bR: \big[Order.max/0]_(i <- l) `|coord B i x| \is Num.real. + exact: bigmax_real. +move: (real_comparable bR (normr_real (coord B j x))). +by move=> /comparable_leNgt <- /(le_trans IHl). +Qed. + +Lemma ler_oo_normD x y : oo_norm (x + y) <= oo_norm x + oo_norm y. +Proof. +apply: bigmax_le => [|/= i _]. + by apply: addr_ge0; apply: oo_norm_ge0. +rewrite raddfD/=. +by apply/(le_trans (ler_normD _ _))/lerD; apply: le_coord_oo_norm. +Qed. + +Lemma oo_norm0_eq0 x : oo_norm x = 0 -> x = 0. +Proof. +move=> x0. +rewrite (coord_basis Bbasis (memvf x)). +suff: forall i, coord B i x = 0. + move=> {}x0. + under eq_bigr do rewrite x0. + by rewrite -scaler_sumr scale0r. +move=> i; apply/normr0_eq0/le_anti/andP; split; last exact: normr_ge0. +by rewrite -x0; apply: le_coord_oo_norm. +Qed. + +Lemma oo_normZ r x : oo_norm (r *: x) = `|r| * oo_norm x. +Proof. +rewrite /oo_norm. +under eq_bigr do rewrite linearZ normrZ/=. +elim: (index_enum _) => [|i l IHl]; first by rewrite !big_nil mulr0. +by rewrite !big_cons IHl maxr_pMr. +Qed. + +Lemma oo_normMn x n : oo_norm (x *+ n) = oo_norm x *+ n. +Proof. by rewrite -scaler_nat oo_normZ normr_nat mulr_natl. Qed. + +Lemma oo_normN x : oo_norm (- x) = oo_norm x. +Proof. by rewrite -scaleN1r oo_normZ normrN1 mul1r. Qed. + +HB.instance Definition _ := Num.Zmodule_isNormed.Build R + oo_space ler_oo_normD oo_norm0_eq0 oo_normMn oo_normN. + +HB.instance Definition _ := + PseudoMetric.copy oo_space (pseudoMetric_normed oo_space). + +HB.instance Definition _ := NormedZmod_PseudoMetric_eq.Build R oo_space erefl. + +HB.instance Definition _ := + PseudoMetricNormedZmod_Lmodule_isNormedModule.Build R oo_space oo_normZ. + +End InfiniteNorm. + +Section EquivalenceNorms. +Variables (R : realType) (V : vectType R). +Let V' := @fullv _ V. +Let Voo := oo_space (vbasisP (@fullv _ V)). + +(* N.B. Get Trocq to prove the continuity part automatically. *) +Lemma oo_closed_ball_compact : compact (closed_ball (0 : Voo) 1). +Proof. +rewrite closed_ballE/closed_ball_//=. +under eq_set do rewrite sub0r normrN. +rewrite -[forall x, _]/(compact _). +pose f (X : {ptws 'I_(\dim V') -> R^o}) : Voo := + \sum_(i < \dim V') (X i) *: (vbasis V')`_i. +have -> : + [set x : Voo | `|x| <= 1] = f @` [set X | forall i, `[-1, 1]%classic (X i)]. + apply/seteqP; split=> x/=. + move=> x1; exists (fun i => coord (vbasis V') i x); last first. + exact/esym/(@coord_vbasis _ _ (x : V))/memvf. + move=> i; rewrite in_itv/= -ler_norml. + apply: (le_trans _ x1). + exact: le_coord_oo_norm. + move=> [X] X1 <-. + rewrite /normr/=/oo_norm. + apply: bigmax_le => //= i _. + rewrite coord_sum_free; last exact/basis_free/vbasisP. + rewrite ler_norml. + exact: X1. +apply: (@continuous_compact _ _ f); last first. + apply: (@tychonoff 'I_(\dim V') (fun=> R^o) (fun=> `[-1, 1 : R^o]%classic)). + move=> _. + exact: segment_compact. +apply/continuous_subspaceT/sum_continuous => i _/= x. +exact/continuousZl/proj_continuous. +Qed. + +Lemma equivalence_norms (N : V -> R) : + N 0 = 0 -> (forall x, 0 <= N x) -> (forall x, N x = 0 -> x = 0) -> + (forall x y, N (x + y) <= N x + N y) -> + (forall r x, N (r *: x) = `|r| * N x) -> + exists M, 0 < M /\ forall x : Voo, `|x| <= M * N x /\ N x <= M * `|x|. +Proof. +move=> N0 N_ge0 N0_eq0 ND NZ. +set M0 := 1 + \sum_(i < \dim V') N (vbasis V')`_i. +have N_sum (I : Type) (r : seq I) (F : I -> V): + N (\sum_(i <- r) F i) <= \sum_(i <- r) N (F i). + elim: r => [|x r IHr]; first by rewrite !big_nil N0. + by rewrite !big_cons; apply/(le_trans (ND _ _))/lerD. +have leNoo: forall x : Voo, N x <= M0 * `|x|. + move=> x. + rewrite [in leLHS](coord_vbasis (memvf (x : V))). + apply: (le_trans (N_sum _ _ _)). + rewrite mulrDl mul1r mulr_suml -[leLHS]add0r. + apply: lerD => //. + apply: ler_sum => /= i _. + rewrite NZ mulrC; apply: ler_pM => //. + exact: le_coord_oo_norm. +have M00 : 0 < M0. + rewrite -[ltLHS]addr0. + by apply: ltr_leD => //; apply: sumr_ge0. +have NC0 : continuous (N : Voo -> R). + move=> /= x. + rewrite /continuous_at. + apply: cvg_zero; first exact: nbhs_filter. + apply/cvgr0Pnorm_le; first exact: nbhs_filter. + move=> /= e e0. + near=> y. + rewrite -[_ y]/(N y - N x). + apply: (@le_trans _ _ (N (y - x))). + apply/real_ler_normlP. + by apply: realB; apply: ger0_real. + have NB a b : N a <= N b + N (a - b). + by rewrite -[a in N a]addr0 -(subrr b) addrCA ND. + rewrite opprB !lerBlDl; split; last exact: NB. + by rewrite -opprB -scaleN1r NZ normrN1 mul1r NB. + apply: (le_trans (leNoo _)). + rewrite mulrC -ler_pdivlMr// -opprB normrN. + near: y; apply: cvgr_dist_le; first exact: cvg_id. + exact: divr_gt0. +have: compact [set x : Voo | `|x| = 1]. + apply: (subclosed_compact _ oo_closed_ball_compact); last first. + apply/subsetP => x. + rewrite closed_ballE// !inE/=. + by rewrite /closed_ball_/= sub0r normrN => ->. + apply: (@closed_comp _ _ _ [set 1 : R]); last exact: closed_eq. + by rewrite /prop_in1 => + _; apply: norm_continuous. +move=> /(@continuous_compact _ _ (N : Voo -> R)) -/(_ _)/wrap[]. + exact: continuous_subspaceT. +move=> /(@continuous_compact _ _ (@GRing.inv R)) -/(_ _)/wrap[]. + move=> /= x. + rewrite /continuous_at. + apply: (@continuous_in_subspaceT _ _ + [set N x | x in [set x : Voo | `|x| = 1]] (@GRing.inv R)). + move=> r /set_mem/=[] y y1 <-. + apply: inv_continuous. + apply/negP => /eqP/N0_eq0 y0. + move: y1; rewrite y0 normr0 => /esym. + by move: (@oner_neq0 R) => /eqP. +move=> /compact_bounded[] M1 [] M1R /(_ (1 + M1)). +rewrite -subr_gt0 -addrA subrr addr0 => /(_ ltr01). +rewrite /globally/= => M1N. +exists (maxr M0 (1 + M1)). +split=> [|x]; first by apply: (lt_le_trans M00); rewrite le_max lexx. +split; last first. + apply/(le_trans (leNoo x))/ler_pM => //; first exact/ltW. + by rewrite /maxr; case: ifP => // /ltW. +have [->|x0] := eqVneq x 0; first by rewrite normr0 N0 mulr0. +have Nx0: 0 < N x. + rewrite lt0r N_ge0 andbT. + by move: x0; apply: contra => /eqP/N0_eq0/eqP. +have normx0 : 0 < `|x|. + move: (lt_le_trans Nx0 (leNoo x)). + by rewrite pmulr_rgt0. +move: M1N => /(_ (`|x| / N x)) -/(_ _)/wrap[]. + exists (N x / `|x|); last by rewrite invf_div. + exists (`|x|^-1 *: x); last first. + by rewrite NZ mulrC ger0_norm. + rewrite normrZ mulrC ger0_norm. + by rewrite divrr//; apply: unitf_gt0. + by rewrite invr_ge0; apply: ltW. +rewrite ger0_norm; last exact: divr_ge0. +rewrite ler_pdivrMr// => xle. +apply: (le_trans xle). +rewrite -subr_ge0 -mulrBl pmulr_lge0// subr_ge0. +by rewrite le_max lexx orbT. +Unshelve. all: by end_near. +Qed. + +End EquivalenceNorms. + +Section LinearContinuous. +Variables (R : realType) (V : normedVectType R) (W : normedModType R). + +Lemma linear_findim_continuous (f : {linear V -> W}) : continuous f. +Proof. +set V' := @fullv _ V. +set B := vbasis V' => /= x. +rewrite /continuous_at. +rewrite [x in f x](coord_vbasis (memvf x)) raddf_sum. +rewrite (@eq_cvg _ _ _ _ (fun y => \sum_(i < \dim V') coord B i y *: f B`_i)); + last first. + move=> y; rewrite [y in LHS](coord_vbasis (memvf y)) raddf_sum. + by apply: eq_big => // i _; apply: linearZ. +apply: cvg_sum => i _. +rewrite [X in _ --> X]linearZ/= -/B. +apply: cvgZl. +move: x; apply/linear_bounded_continuous/bounded_funP => r/=. +have := @equivalence_norms R V (@normr R V) (@normr0 _ _) (@normr_ge0 _ _) + (@normr0_eq0 _ _) (@ler_normD _ _) (@normrZ _ _). +move=> [] M [] M0 MP. +exists (M * r) => x. +move: MP => /(_ x)[] Mx Mx' xr. +apply/(le_trans (le_coord_oo_norm _ _ _))/(le_trans Mx). +rewrite -subr_ge0 -mulrBr; apply: mulr_ge0; first exact: ltW. +by rewrite subr_ge0. +Qed. + +End LinearContinuous. + diff --git a/theories/tvs.v b/theories/tvs.v index fe43593043..46dfb04531 100644 --- a/theories/tvs.v +++ b/theories/tvs.v @@ -90,6 +90,22 @@ HB.mixin Record PreTopologicalNmodule_isTopologicalNmodule M HB.structure Definition TopologicalNmodule := {M of PreTopologicalNmodule M & PreTopologicalNmodule_isTopologicalNmodule M}. +Section TopologicalNmodule_Theory. + +Lemma cvg_sum (T : Type) (U : TopologicalNmodule.type) (F : set_system T) + (I : Type) (r : seq I) (P : pred I) (Ff : I -> T -> U) (Fa : I -> U) : + Filter F -> (forall i, P i -> Ff i x @[x --> F] --> Fa i) -> + \sum_(i <- r | P i) Ff i x @[x --> F] --> \sum_(i <- r| P i) Fa i. +Proof. by move=> FF Ffa; apply: cvg_big => //; apply: add_continuous. Qed. + +Lemma sum_continuous (T : topologicalType) (M : TopologicalNmodule.type) + (I : Type) (r : seq I) (P : pred I) (F : I -> T -> M) : + (forall i : I, P i -> continuous (F i)) -> + continuous (fun x1 : T => \sum_(i <- r | P i) F i x1). +Proof. by move=> FC0; apply: continuous_big => //; apply: add_continuous. Qed. + +End TopologicalNmodule_Theory. + HB.structure Definition PreTopologicalZmodule := {M of Topological M & GRing.Zmodule M}. From 99d840c0404b054c2be4cfac217891a4064e7521 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 8 Dec 2025 10:15:16 +0900 Subject: [PATCH 2/2] linting --- theories/normedtype_theory/normed_module.v | 193 ++++++++------------- 1 file changed, 71 insertions(+), 122 deletions(-) diff --git a/theories/normedtype_theory/normed_module.v b/theories/normedtype_theory/normed_module.v index 1ac0300a38..3e6e79330c 100644 --- a/theories/normedtype_theory/normed_module.v +++ b/theories/normedtype_theory/normed_module.v @@ -2088,9 +2088,10 @@ Qed. End Closed_Ball_normedModType. Section InfiniteNorm. -Variables (R : numFieldType) (V : vectType R) (B : (\dim (@fullv _ V)).-tuple V). +Variables (R : numFieldType) (V : vectType R). Let V' := @fullv _ V. -Hypothesis (Bbasis : basis_of V' B). +Variable B : (\dim V').-tuple V. +Hypothesis Bbasis : basis_of V' B. Definition oo_norm x := \big[Order.max/0]_(i < \dim V') `|coord B i x|. @@ -2103,40 +2104,31 @@ HB.instance Definition _ := Pointed.copy oo_space V^o. Lemma oo_norm_ge0 x : 0 <= oo_norm x. Proof. rewrite /oo_norm. -elim: (index_enum _) => /=; first by rewrite big_nil. -by move=> i l IHl; rewrite big_cons /Order.max/=; case: ifP. +by elim/big_ind : _ => //= ? ? ? ?; rewrite /Order.max; case: ifP. Qed. Lemma le_coord_oo_norm x i : `|coord B i x| <= oo_norm x. Proof. rewrite /oo_norm; elim: (index_enum _) (mem_index_enum i) => //= j l IHl. -rewrite inE big_cons /Order.max/= => /orP[/eqP <-|/IHl {}IHl]; - case: ifP => [/ltW|]//. -move=> /negP/negP. -have bR: \big[Order.max/0]_(i <- l) `|coord B i x| \is Num.real. - exact: bigmax_real. -move: (real_comparable bR (normr_real (coord B j x))). -by move=> /comparable_leNgt <- /(le_trans IHl). +rewrite inE big_cons [X in _ <= X _ _]/Order.max/= => /predU1P[<-|/IHl {}IHl]; + case: ifP => [/ltW|]// /negbT. +set b := (X in _ < X); have bR : b \is Num.real by exact: bigmax_real. +have /comparable_leNgt <- := real_comparable bR (normr_real (coord B j x)). +by move=> /(le_trans IHl). Qed. Lemma ler_oo_normD x y : oo_norm (x + y) <= oo_norm x + oo_norm y. Proof. -apply: bigmax_le => [|/= i _]. - by apply: addr_ge0; apply: oo_norm_ge0. -rewrite raddfD/=. -by apply/(le_trans (ler_normD _ _))/lerD; apply: le_coord_oo_norm. +apply: bigmax_le => [|/= i _]; first by rewrite addr_ge0// oo_norm_ge0. +by rewrite raddfD/= (le_trans (ler_normD _ _))// lerD// le_coord_oo_norm. Qed. Lemma oo_norm0_eq0 x : oo_norm x = 0 -> x = 0. Proof. -move=> x0. -rewrite (coord_basis Bbasis (memvf x)). +move=> x0; rewrite (coord_basis Bbasis (memvf x)). suff: forall i, coord B i x = 0. - move=> {}x0. - under eq_bigr do rewrite x0. - by rewrite -scaler_sumr scale0r. -move=> i; apply/normr0_eq0/le_anti/andP; split; last exact: normr_ge0. -by rewrite -x0; apply: le_coord_oo_norm. + by move=> {}x0; rewrite big1//= => j _; rewrite x0 scale0r. +by move=> i; apply/normr0_eq0/le_anti; rewrite normr_ge0 -x0 le_coord_oo_norm. Qed. Lemma oo_normZ r x : oo_norm (r *: x) = `|r| * oo_norm x. @@ -2174,124 +2166,85 @@ Let Voo := oo_space (vbasisP (@fullv _ V)). (* N.B. Get Trocq to prove the continuity part automatically. *) Lemma oo_closed_ball_compact : compact (closed_ball (0 : Voo) 1). Proof. -rewrite closed_ballE/closed_ball_//=. +rewrite closed_ballE// /closed_ball_. under eq_set do rewrite sub0r normrN. rewrite -[forall x, _]/(compact _). -pose f (X : {ptws 'I_(\dim V') -> R^o}) : Voo := - \sum_(i < \dim V') (X i) *: (vbasis V')`_i. -have -> : - [set x : Voo | `|x| <= 1] = f @` [set X | forall i, `[-1, 1]%classic (X i)]. - apply/seteqP; split=> x/=. - move=> x1; exists (fun i => coord (vbasis V') i x); last first. +pose f (X : {ptws 'I_(\dim V') -> R}) : Voo := + \sum_(i < \dim V') X i *: (vbasis V')`_i. +have -> : [set x : Voo | `|x| <= 1] = + f @` [set X | forall i, `[-1, 1]%classic (X i)]. + apply/seteqP; split=> [x/= x1|x/= [X X1 <-]]. + - exists (coord (vbasis V') ^~ x); last first. exact/esym/(@coord_vbasis _ _ (x : V))/memvf. - move=> i; rewrite in_itv/= -ler_norml. - apply: (le_trans _ x1). - exact: le_coord_oo_norm. - move=> [X] X1 <-. - rewrite /normr/=/oo_norm. - apply: bigmax_le => //= i _. - rewrite coord_sum_free; last exact/basis_free/vbasisP. - rewrite ler_norml. - exact: X1. -apply: (@continuous_compact _ _ f); last first. - apply: (@tychonoff 'I_(\dim V') (fun=> R^o) (fun=> `[-1, 1 : R^o]%classic)). - move=> _. + by move=> i; rewrite in_itv/= -ler_norml (le_trans _ x1)// le_coord_oo_norm. + - rewrite /normr/= /oo_norm bigmax_le => //= i _. + by rewrite coord_sum_free ?ler_norml; [exact: X1|exact/basis_free/vbasisP]. +apply: (@continuous_compact _ _ f). +- apply/continuous_subspaceT/sum_continuous => /= i _ x. + exact/continuousZr_tmp/proj_continuous. +- apply: (@tychonoff 'I_(\dim V') (fun=> R^o) (fun=> `[-1, 1]%classic)) => _. exact: segment_compact. -apply/continuous_subspaceT/sum_continuous => i _/= x. -exact/continuousZl/proj_continuous. Qed. Lemma equivalence_norms (N : V -> R) : N 0 = 0 -> (forall x, 0 <= N x) -> (forall x, N x = 0 -> x = 0) -> (forall x y, N (x + y) <= N x + N y) -> (forall r x, N (r *: x) = `|r| * N x) -> - exists M, 0 < M /\ forall x : Voo, `|x| <= M * N x /\ N x <= M * `|x|. + exists2 M, 0 < M & forall x : Voo, `|x| <= M * N x /\ N x <= M * `|x|. Proof. move=> N0 N_ge0 N0_eq0 ND NZ. set M0 := 1 + \sum_(i < \dim V') N (vbasis V')`_i. -have N_sum (I : Type) (r : seq I) (F : I -> V): - N (\sum_(i <- r) F i) <= \sum_(i <- r) N (F i). - elim: r => [|x r IHr]; first by rewrite !big_nil N0. - by rewrite !big_cons; apply/(le_trans (ND _ _))/lerD. -have leNoo: forall x : Voo, N x <= M0 * `|x|. - move=> x. - rewrite [in leLHS](coord_vbasis (memvf (x : V))). - apply: (le_trans (N_sum _ _ _)). - rewrite mulrDl mul1r mulr_suml -[leLHS]add0r. - apply: lerD => //. - apply: ler_sum => /= i _. - rewrite NZ mulrC; apply: ler_pM => //. - exact: le_coord_oo_norm. -have M00 : 0 < M0. - rewrite -[ltLHS]addr0. - by apply: ltr_leD => //; apply: sumr_ge0. +have M00 : 0 < M0 by rewrite ltr_pwDl// sumr_ge0. +have N_sum (I : Type) (r : seq I) (F : I -> V) : + N (\sum_(i <- r) F i) <= \sum_(i <- r) N (F i). + by elim/big_ind2 : _ => *; rewrite ?N0// (le_trans (ND _ _))// lerD. +have leNoo (x : Voo) : N x <= M0 * `|x|. + rewrite [in leLHS](coord_vbasis (memvf (x : V))) (le_trans (N_sum _ _ _))//. + rewrite mulrDl mul1r mulr_suml ler_wpDl// ler_sum => //= i _. + by rewrite NZ mulrC ler_pM// le_coord_oo_norm. +have NZN a : N (- a) = N a by rewrite -scaleN1r NZ normrN1 mul1r. have NC0 : continuous (N : Voo -> R). - move=> /= x. - rewrite /continuous_at. + move=> /= x; rewrite /continuous_at. apply: cvg_zero; first exact: nbhs_filter. apply/cvgr0Pnorm_le; first exact: nbhs_filter. move=> /= e e0. near=> y. - rewrite -[_ y]/(N y - N x). - apply: (@le_trans _ _ (N (y - x))). - apply/real_ler_normlP. - by apply: realB; apply: ger0_real. - have NB a b : N a <= N b + N (a - b). - by rewrite -[a in N a]addr0 -(subrr b) addrCA ND. - rewrite opprB !lerBlDl; split; last exact: NB. - by rewrite -opprB -scaleN1r NZ normrN1 mul1r NB. - apply: (le_trans (leNoo _)). - rewrite mulrC -ler_pdivlMr// -opprB normrN. - near: y; apply: cvgr_dist_le; first exact: cvg_id. - exact: divr_gt0. + rewrite -[_ y]/(N y - N x) (@le_trans _ _ (N (y - x)))//. + apply/ler_normlP. + have NB a b : N a <= N b + N (a - b). + by rewrite (le_trans _ (ND _ _)) ?subrKC. + by rewrite opprB !lerBlDl NB -opprB NZN NB. + rewrite (le_trans (leNoo _))// mulrC -ler_pdivlMr// -opprB normrN. + by near: y; apply: cvgr_dist_le; [exact: cvg_id|exact: divr_gt0]. have: compact [set x : Voo | `|x| = 1]. - apply: (subclosed_compact _ oo_closed_ball_compact); last first. - apply/subsetP => x. - rewrite closed_ballE// !inE/=. - by rewrite /closed_ball_/= sub0r normrN => ->. - apply: (@closed_comp _ _ _ [set 1 : R]); last exact: closed_eq. - by rewrite /prop_in1 => + _; apply: norm_continuous. + apply: (subclosed_compact _ oo_closed_ball_compact). + - apply: (@closed_comp _ _ _ [set 1 : R]); last exact: closed_eq. + by move=> *; exact: norm_continuous. + - by move => x/=; rewrite closed_ballE// /closed_ball_/= sub0r normrN => ->. move=> /(@continuous_compact _ _ (N : Voo -> R)) -/(_ _)/wrap[]. exact: continuous_subspaceT. move=> /(@continuous_compact _ _ (@GRing.inv R)) -/(_ _)/wrap[]. - move=> /= x. - rewrite /continuous_at. + move=> /= x; rewrite /continuous_at. apply: (@continuous_in_subspaceT _ _ [set N x | x in [set x : Voo | `|x| = 1]] (@GRing.inv R)). - move=> r /set_mem/=[] y y1 <-. + move=> /= r /set_mem/= [y y1 <-]. apply: inv_continuous. - apply/negP => /eqP/N0_eq0 y0. - move: y1; rewrite y0 normr0 => /esym. - by move: (@oner_neq0 R) => /eqP. -move=> /compact_bounded[] M1 [] M1R /(_ (1 + M1)). -rewrite -subr_gt0 -addrA subrr addr0 => /(_ ltr01). + by apply: contra_eq_neq y1 => /N0_eq0 ->; rewrite normr0 eq_sym oner_eq0. +move=> /compact_bounded[M1 [M1R /(_ (1 + M1))]] /(_ (ltr_pwDl ltr01 (lexx _))). rewrite /globally/= => M1N. -exists (maxr M0 (1 + M1)). -split=> [|x]; first by apply: (lt_le_trans M00); rewrite le_max lexx. -split; last first. - apply/(le_trans (leNoo x))/ler_pM => //; first exact/ltW. - by rewrite /maxr; case: ifP => // /ltW. +exists (maxr M0 (1 + M1)) => [|x]; first by rewrite lt_max M00. +split; last by rewrite (le_trans (leNoo x))// ler_wpM2r// le_max lexx. have [->|x0] := eqVneq x 0; first by rewrite normr0 N0 mulr0. -have Nx0: 0 < N x. - rewrite lt0r N_ge0 andbT. - by move: x0; apply: contra => /eqP/N0_eq0/eqP. -have normx0 : 0 < `|x|. - move: (lt_le_trans Nx0 (leNoo x)). - by rewrite pmulr_rgt0. +have Nx0 : 0 < N x. + by rewrite lt0r N_ge0 andbT; move: x0; apply: contra_neq => /N0_eq0. +have normx0 : 0 < `|x| by rewrite normr_gt0. move: M1N => /(_ (`|x| / N x)) -/(_ _)/wrap[]. exists (N x / `|x|); last by rewrite invf_div. - exists (`|x|^-1 *: x); last first. - by rewrite NZ mulrC ger0_norm. - rewrite normrZ mulrC ger0_norm. - by rewrite divrr//; apply: unitf_gt0. - by rewrite invr_ge0; apply: ltW. -rewrite ger0_norm; last exact: divr_ge0. -rewrite ler_pdivrMr// => xle. -apply: (le_trans xle). -rewrite -subr_ge0 -mulrBl pmulr_lge0// subr_ge0. -by rewrite le_max lexx orbT. -Unshelve. all: by end_near. -Qed. + exists (`|x|^-1 *: x); last by rewrite NZ mulrC ger0_norm. + by rewrite normrZ normfV normr_id mulVf// gt_eqF. +rewrite ger0_norm ?divr_ge0// ler_pdivrMr// => /le_trans; apply. +by rewrite ler_pM2r// le_max lexx orbT. +Unshelve. all: by end_near. Qed. End EquivalenceNorms. @@ -2301,8 +2254,8 @@ Variables (R : realType) (V : normedVectType R) (W : normedModType R). Lemma linear_findim_continuous (f : {linear V -> W}) : continuous f. Proof. set V' := @fullv _ V. -set B := vbasis V' => /= x. -rewrite /continuous_at. +set B := vbasis V'. +move=> /= x; rewrite /continuous_at. rewrite [x in f x](coord_vbasis (memvf x)) raddf_sum. rewrite (@eq_cvg _ _ _ _ (fun y => \sum_(i < \dim V') coord B i y *: f B`_i)); last first. @@ -2310,17 +2263,13 @@ rewrite (@eq_cvg _ _ _ _ (fun y => \sum_(i < \dim V') coord B i y *: f B`_i)); by apply: eq_big => // i _; apply: linearZ. apply: cvg_sum => i _. rewrite [X in _ --> X]linearZ/= -/B. -apply: cvgZl. +apply: cvgZr_tmp. move: x; apply/linear_bounded_continuous/bounded_funP => r/=. -have := @equivalence_norms R V (@normr R V) (@normr0 _ _) (@normr_ge0 _ _) - (@normr0_eq0 _ _) (@ler_normD _ _) (@normrZ _ _). -move=> [] M [] M0 MP. +have [M M0 MP] := @equivalence_norms R V (@normr R V) (@normr0 _ _) + (@normr_ge0 _ _) (@normr0_eq0 _ _) (@ler_normD _ _) (@normrZ _ _). exists (M * r) => x. -move: MP => /(_ x)[] Mx Mx' xr. -apply/(le_trans (le_coord_oo_norm _ _ _))/(le_trans Mx). -rewrite -subr_ge0 -mulrBr; apply: mulr_ge0; first exact: ltW. -by rewrite subr_ge0. +move: MP => /(_ x) [Mx _] xr. +by rewrite (le_trans (le_coord_oo_norm _ _ _))// (le_trans Mx) ?ler_wpM2l// ltW. Qed. End LinearContinuous. -