| import OAI.Geometry.NodalSets.Spectral.FiniteEigenframeMinMax | |
| namespace OAI | |
| namespace Yau.Analysis | |
| open Set Metric | |
| noncomputable section | |
| theorem compact_eigenframe_card_bound {E : Type*} [NormedAddCommGroup E] | |
| [InnerProductSpace ℝ E] (T : E →L[ℝ] E) (hT : IsCompactOperator T) | |
| (c : ℝ) (hc : 0 < c) : | |
| ∃ K : ℕ, ∀ (N : ℕ) (e : Fin N → E) (mu : Fin N → ℝ), | |
| Orthonormal ℝ e → (∀ i, T (e i)=mu i • e i) → | |
| (∀ i, c ≤ mu i) → N ≤ K := by | |
| classical | |
| have hcomp := hT.isCompact_closure_image_closedBall (f := T.toLinearMap) 1 | |
| obtain ⟨t,ht,hcover⟩ := Metric.totallyBounded_iff.mp hcomp.totallyBounded (c/2) (half_pos hc) | |
| let := ht.fintype | |
| refine ⟨Fintype.card t,?_⟩ | |
| intro N e mu ho he hmu | |
| have hnear (i : Fin N) : ∃ a : t, dist (T (e i)) a < c/2 := by | |
| have hx : T (e i) ∈ closure (T '' closedBall 0 1) := | |
| subset_closure ⟨e i,by simpa using (ho.norm_eq_one i).le,rfl⟩ | |
| have hxcover := hcover hx | |
| simp only [mem_iUnion,mem_ball] at hxcover | |
| obtain ⟨a,ha,hai⟩ := hxcover | |
| exact ⟨⟨a,ha⟩,hai⟩ | |
| choose f hf using hnear | |
| have hinj : Function.Injective f := by | |
| intro i j hij | |
| by_contra hne | |
| have hinner : inner ℝ (e i) (T (e i)-T (e j)) = mu i := by | |
| rw [he i,he j,inner_sub_right,inner_smul_right,inner_smul_right, | |
| real_inner_self_eq_norm_sq,ho.norm_eq_one i,ho.inner_eq_zero hne] | |
| simp | |
| have hdist := real_inner_le_norm (e i) (T (e i)-T (e j)) | |
| rw [ho.norm_eq_one i,one_mul,hinner,← dist_eq_norm] at hdist | |
| have hd : dist (T (e i)) (T (e j)) < c := by | |
| have h1 := hf i | |
| have h2 := hf j | |
| rw [← hij] at h2 | |
| have hh := dist_triangle (T (e i)) (f i : E) (T (e j)) | |
| rw [dist_comm (f i : E) (T (e j))] at hh | |
| linarith | |
| exact (not_lt_of_ge ((hmu i).trans hdist)) hd | |
| simpa using Fintype.card_le_of_injective f hinj | |
| end | |
| end Yau.Analysis | |
| end OAI | |