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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import Mathlib
import OAI.Analysis.Commutator.HilbertSequence
import OAI.Analysis.Commutator.UltrafilterDiagonal
namespace OAI
noncomputable section
section
noncomputable section
open scoped Topology InnerProductSpace BoundedContinuousFunction
open Filter UniformSpace Completion
namespace Kadison.HilbertSequence
variable {A H : Type*} [NormedAddCommGroup A] [NormedSpace ℂ A]
[NormedAddCommGroup H] [InnerProductSpace ℂ H]
variable (ω : Ultrafilter ℕ) (v : A →L[ℂ] H)
def mapPre (a : ℕ →ᵇ A) : Pre ω H :=
⟨BoundedContinuousFunction.comp v v.lipschitzWith a⟩
def mapVector (a : ℕ →ᵇ A) : Space ω H := (mapPre ω v a : Space ω H)
omit [InnerProductSpace ℂ H] in
@[simp] lemma seq_sub' (a b : Pre ω H) (i : ℕ) : (a-b).seq i=a.seq i-b.seq i := rfl
lemma norm_map_difference_tendsto (a b : ℕ →ᵇ A) :
Tendsto (fun i => ‖v (a i)-v (b i)‖) (ω : Filter ℕ)
(𝓝 ‖mapVector ω v a-mapVector ω v b‖) := by
simpa only [mapVector,←Completion.coe_sub,Completion.norm_coe,seq_sub',mapPre,BoundedContinuousFunction.comp_apply] using
tendsto_norm ω H (mapPre ω v a-mapPre ω v b)
theorem bounded_image_isClosed (hω : (ω : Filter ℕ)≤atTop)
(K : Set A) (C : ℝ) (hK : ∀ a∈K, ‖a‖≤C) :
IsClosed {x : Space ω H | ∃ a : ℕ →ᵇ A, (∀ i, a i∈K) ∧ mapVector ω v a=x} := by
classical
apply isClosed_of_closure_subset
intro x hx
let ε (n : ℕ) : ℝ := (1/2:ℝ)^n
have hε (n : ℕ) : 0<ε n := pow_pos (by norm_num) n
have hanti {j k : ℕ} (hjk : j≤k) : ε k≤ε j :=
pow_le_pow_of_le_one (by norm_num) (by norm_num) hjk
have hn (n : ℕ) : ∃ a : ℕ →ᵇ A, (∀ i, a i∈K) ∧ ‖mapVector ω v a-x‖<ε n := by
obtain ⟨y,⟨a,ha,rfl⟩,hy⟩ := Metric.mem_closure_iff.mp hx (ε n) (hε n)
exact ⟨a,ha,by simpa only [dist_eq_norm,norm_sub_rev] using hy⟩
choose f hf he using hn
have hpair (j k : ℕ) (hjk : j≤k) :
∀ᶠ i in (ω : Filter ℕ), ‖v (f j i)-v (f k i)‖≤3*ε j := by
have hlt : ‖mapVector ω v (f j)-mapVector ω v (f k)‖<3*ε j := by
have h := norm_sub_le_norm_sub_add_norm_sub (mapVector ω v (f j)) x (mapVector ω v (f k))
rw [norm_sub_rev x] at h
have hj := he j
have hk := he k
have hk' := hanti hjk
have hp := hε j
linarith
exact (norm_map_difference_tendsto ω v (f j) (f k)).eventually (gt_mem_nhds hlt) |>.mono
(fun _ hi => le_of_lt hi)
let p (n i : ℕ) : Prop := ∀ j≤n, ∀ k≤n, j≤k → ‖v (f j i)-v (f k i)‖≤3*ε j
have hp (n : ℕ) : ∀ᶠ i in (ω : Filter ℕ), p n i := by
have hh : ∀ᶠ i in (ω : Filter ℕ), ∀ j∈Finset.range (n+1),
∀ k∈Finset.range (n+1), j≤k → ‖v (f j i)-v (f k i)‖≤3*ε j := by
apply (eventually_all_finset (Finset.range (n+1))).mpr
intro j _
apply (eventually_all_finset (Finset.range (n+1))).mpr
intro k _
by_cases hjk : j≤k
· exact (hpair j k hjk).mono (fun _ hi _ => hi)
· exact Eventually.of_forall (fun _ hi => (hjk hi).elim)
filter_upwards [hh] with i hi
intro j hj k hk hjk
exact hi j (Finset.mem_range.mpr (by omega)) k (Finset.mem_range.mpr (by omega)) hjk
obtain ⟨d,hd,hreq⟩ := UltrafilterDiagonal.depth hω p hp
let a : ℕ →ᵇ A := BoundedContinuousFunction.mkOfDiscrete (fun i => f (d i/2) i) (2*C)
(fun i j => (dist_le_norm_add_norm _ _).trans (by
have hi := hK _ (hf (d i/2) i)
have hj := hK _ (hf (d j/2) j)
linarith))
have ha (i : ℕ) : a i∈K := hf (d i/2) i
have hnorm (j : ℕ) : ‖mapVector ω v a-mapVector ω v (f j)‖≤3*ε j := by
apply le_of_tendsto (norm_map_difference_tendsto ω v a (f j))
filter_upwards [hd.eventually (eventually_ge_atTop (2*j+2))] with i hi
have hleft : j≤d i/2 := by omega
have hright : d i/2<d i := by omega
have hh := hreq i (d i/2) hright j hleft (d i/2) (le_refl _) hleft
simpa only [a,BoundedContinuousFunction.mkOfDiscrete_apply,norm_sub_rev] using hh
have hnorm' (j : ℕ) : ‖mapVector ω v a-x‖≤4*ε j := by
have h := norm_sub_le_norm_sub_add_norm_sub (mapVector ω v a) (mapVector ω v (f j)) x
have h1 := hnorm j
have h2 := he j
linarith
have hlim : Tendsto (fun j => 4*ε j) atTop (𝓝 (0:ℝ)) := by
simpa only [mul_zero] using
(tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_zero_of_lt_one
(by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)<1)))
have hz : ‖mapVector ω v a-x‖≤0 := ge_of_tendsto hlim (Eventually.of_forall hnorm')
exact ⟨a,ha,sub_eq_zero.mp (norm_le_zero_iff.mp hz)⟩
end Kadison.HilbertSequence
end
end
end
end OAI
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