| import OAI.Analysis.FreeEntropy.MatrixBilinear |
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| namespace OAI |
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| universe u1 |
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| noncomputable section |
| open scoped ComplexConjugate |
| open Finset Metric Set |
| namespace FiniteEntropySeparation |
| variable {E : Type u1} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| theorem norm_le_of_unit_inner (z : E) {C : ℝ} (hC : 0 ≤ C) |
| (h : ∀ v : E, ‖v‖ = 1 → ‖inner ℂ v z‖ ≤ C) : ‖z‖ ≤ C := by |
| by_cases hz : z=0 |
| · simpa [hz] using hC |
| have hn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz |
| have hv : ‖(‖z‖⁻¹ : ℝ) • z‖ = 1 := by rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, inv_mul_cancel₀ hn] |
| have hh := h ((‖z‖⁻¹ : ℝ) • z) hv |
| simpa [inner_smul_real_left, norm_smul, inner_self_eq_norm_sq_to_K, norm_pow, |
| Real.norm_eq_abs, abs_inv, abs_norm, sq, ← mul_assoc, inv_mul_cancel₀ hn] using hh |
| theorem opNorm_le_of_bilinear (T : E →L[ℂ] E) {C : ℝ} (hC : 0 ≤ C) |
| (h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖inner ℂ x (T y)‖ ≤ C) : ‖T‖ ≤ C := by |
| apply ContinuousLinearMap.opNorm_le_of_unit_norm hC |
| intro y hy |
| exact norm_le_of_unit_inner _ hC (fun x hx => h x y hx hy) |
| theorem bilinear_net_bound (T : E →L[ℂ] E) (s : Finset E) |
| (hs : ∀ x ∈ s, ‖x‖ = 1) |
| (hcov : ∀ x : E, ‖x‖ = 1 → ∃ y ∈ s, dist x y ≤ 1/4) |
| {M : ℝ} (hM : 0 ≤ M) (hB : ∀ x ∈ s, ∀ y ∈ s, ‖inner ℂ x (T y)‖ ≤ M) : ‖T‖ ≤ 2*M := by |
| have hb : ‖T‖ ≤ M + ‖T‖/2 := by |
| apply opNorm_le_of_bilinear T (by positivity) |
| intro x y hx hy |
| obtain ⟨u,hu,hxu⟩ := hcov x hx |
| obtain ⟨v,hv,hyv⟩ := hcov y hy |
| have he : inner ℂ x (T y) = inner ℂ u (T v) + inner ℂ (x-u) (T y) + inner ℂ u (T (y-v)) := by |
| simp only [map_sub, inner_sub_left, inner_sub_right]; abel |
| have h1 : ‖inner ℂ (x-u) (T y)‖ ≤ ‖T‖/4 := by |
| calc |
| _ ≤ ‖x-u‖*‖T y‖ := norm_inner_le_norm _ _ |
| _ ≤ (1/4)*(‖T‖*‖y‖) := mul_le_mul (by simpa [dist_eq_norm] using hxu) (T.le_opNorm y) (norm_nonneg _) (by norm_num) |
| _ = _ := by rw [hy]; ring |
| have h2 : ‖inner ℂ u (T (y-v))‖ ≤ ‖T‖/4 := by |
| calc |
| _ ≤ ‖u‖*‖T (y-v)‖ := norm_inner_le_norm _ _ |
| _ ≤ ‖u‖*(‖T‖*‖y-v‖) := mul_le_mul_of_nonneg_left (T.le_opNorm _) (norm_nonneg _) |
| _ ≤ 1*(‖T‖*(1/4)) := by rw [hs u hu]; gcongr; simpa [dist_eq_norm] using hyv |
| _ = _ := by ring |
| rw [he] |
| exact ((norm_add_le _ _).trans (add_le_add ((norm_add_le _ _).trans (add_le_add (hB u hu v hv) h1)) h2)).trans_eq (by ring) |
| linarith |
| theorem sphere_finite_net [FiniteDimensional ℝ E] : |
| ∃ s : Finset E, (∀ x ∈ s, ‖x‖ = 1) ∧ |
| (∀ x : E, ‖x‖ = 1 → ∃ y ∈ s, dist x y ≤ 1/4) ∧ |
| (s.card : ℝ) ≤ 9^Module.finrank ℝ E := by |
| obtain ⟨s,hs,hc,hb⟩ := compact_finite_net (sphere (0:E) 1) (isCompact_sphere 0 1) |
| (R := 1) (ε := 1/4) (by norm_num) (by norm_num) (by intro x hx; exact (mem_sphere_zero_iff_norm.mp hx).le) |
| refine ⟨s,?_,?_,?_⟩ |
| · intro x hx; exact mem_sphere_zero_iff_norm.mp (hs x hx) |
| · intro x hx; exact hc x (mem_sphere_zero_iff_norm.mpr hx) |
| · norm_num at hb ⊢; exact hb |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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