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8fadf68 | 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 | 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
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