| import Mathlib |
| import OAI.Analysis.Commutator.HilbertSequence |
| import OAI.Analysis.Commutator.UltrafilterDiagonal |
|
|
| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open scoped Topology InnerProductSpace BoundedContinuousFunction |
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| open Filter UniformSpace Completion |
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| namespace Kadison.HilbertSequence |
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| variable {A H : Type*} [NormedAddCommGroup A] [NormedSpace ℂ A] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
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| variable (ω : Ultrafilter ℕ) (v : A →L[ℂ] H) |
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| def mapPre (a : ℕ →ᵇ A) : Pre ω H := |
| ⟨BoundedContinuousFunction.comp v v.lipschitzWith a⟩ |
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| def mapVector (a : ℕ →ᵇ A) : Space ω H := (mapPre ω v a : Space ω H) |
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| 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 |
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| 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) |
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| 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 |
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| end |
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| end |
| end |
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| end OAI |
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