File size: 3,019 Bytes
785c727 | 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 67 68 69 70 71 | import Mathlib.Analysis.Normed.Operator.Compact.Basic
namespace OAI
/-! Compactness passes to a linear map whose norm is controlled by a compact
map. This is used for the finite coordinate reconstruction of two-forms. -/
namespace PositiveEinsteinFour.Geometry
open Set Metric
noncomputable section
theorem compact_of_norm_domination
{E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
(T : E →L[ℝ] F) (Q : E →L[ℝ] G) (hQ : IsCompactOperator Q)
(C : ℝ) (hC : 0 ≤ C) (hb : ∀ x, ‖T x‖ ≤ C * ‖Q x‖) :
IsCompactOperator T := by
let R := Set.range (Q : E → G)
let pick : R → E := fun y => Classical.choose y.property
have hpick (y : R) : Q (pick y) = y.val := Classical.choose_spec y.property
let f : R → F := fun y => T (pick y)
have hf (x : E) : f ⟨Q x, ⟨x, rfl⟩⟩ = T x := by
have h := hb (pick ⟨Q x, ⟨x, rfl⟩⟩ - x)
simp only [map_sub, hpick, sub_self, norm_zero, mul_zero] at h
exact sub_eq_zero.mp (norm_le_zero_iff.mp h)
have hLip : LipschitzWith ⟨C, hC⟩ f := by
apply LipschitzWith.of_dist_le_mul
intro x y
change dist (T (pick x)) (T (pick y)) ≤ C * dist x.val y.val
simp only [dist_eq_norm]
simpa only [map_sub, hpick] using hb (pick x - pick y)
let B := closedBall (0 : E) 1
have ht : TotallyBounded (Q '' B) :=
(hQ.isCompact_closure_image_closedBall 1).totallyBounded.subset subset_closure
have hs : TotallyBounded ((Subtype.val : R → G) ⁻¹' (Q '' B)) :=
totallyBounded_preimage isUniformEmbedding_subtype_val.isUniformInducing ht
have heq : f '' ((Subtype.val : R → G) ⁻¹' (Q '' B)) = T '' B := by
ext z
constructor
· rintro ⟨y, ⟨x, hx, hxy⟩, rfl⟩
refine ⟨x, hx, ?_⟩
have hy : y = ⟨Q x, ⟨x, rfl⟩⟩ := Subtype.ext hxy.symm
rw [hy, hf]
· rintro ⟨x, hx, rfl⟩
exact ⟨⟨Q x, ⟨x, rfl⟩⟩, ⟨x, hx, rfl⟩, hf x⟩
have htb : TotallyBounded (T '' B) := heq ▸ hs.image hLip.uniformContinuous
exact (isCompactOperator_iff_isCompact_closure_image_closedBall
T.toLinearMap (show (0 : ℝ) < 1 by norm_num)).mpr
(isCompact_iff_totallyBounded_isComplete.mpr
⟨htb.closure, isClosed_closure.isComplete⟩)
theorem compact_finite_pi
{E F J : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [Fintype J]
(Q : J → E →L[ℝ] F) (hQ : ∀ j, IsCompactOperator (Q j)) :
IsCompactOperator (ContinuousLinearMap.pi Q) := by
choose K hK hb using fun j => (hQ j).image_closedBall_subset_compact 1
apply (isCompactOperator_iff_image_closedBall_subset_compact
(ContinuousLinearMap.pi Q).toLinearMap (show (0 : ℝ) < 1 by norm_num)).mpr
refine ⟨Set.pi Set.univ K, isCompact_univ_pi hK, ?_⟩
rintro _ ⟨x, hx, rfl⟩ j _
exact hb j ⟨x, hx, rfl⟩
end
end PositiveEinsteinFour.Geometry
end OAI
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