Normed vector types, infinite norm, norm equivalence thm, continuity …

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`:

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 +2086,190 @@ by rewrite interior_closed_ballE //; exact: ballxx.
 Qed.
 
 End Closed_Ball_normedModType.
+
+Section InfiniteNorm.
+Variables (R : numFieldType) (V : vectType R).
+Let V' := @fullv _ V.
+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|.
+
+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.
+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 [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 _]; 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)).
+suff: forall i, coord B i x = 0.
+  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.
+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}) : 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.
+    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.
+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) ->
+  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 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.
+  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) (@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).
+  - 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.
+  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.
+  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)) => [|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.
+  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 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.
+
+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'.
+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.
+  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: cvgZr_tmp.
+move: x; apply/linear_bounded_continuous/bounded_funP => r/=.
+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 _] xr.
+by rewrite (le_trans (le_coord_oo_norm _ _ _))// (le_trans Mx) ?ler_wpM2l// ltW.
+Qed.
+
+End LinearContinuous.

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}.