File size: 3,268 Bytes
67aee1d | 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 | import OAI.Analysis.FreeEntropy.MatrixBilinear
namespace OAI
universe u1
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
end
end OAI
|