| import Mathlib.Analysis.Normed.Operator.Compact.Basic |
|
|
| namespace OAI |
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|
| /-! 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 |
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|
| open Set Metric |
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|
| noncomputable section |
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|
| 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 |
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|
| end OAI |
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|
