| import OAI.Analysis.Daugavet.ExactModuli |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal NNReal Topology |
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| universe uX |
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| namespace ExactModuli |
| variable {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| lemma normalized_functional_slice_nonempty (ell : X →L[ℝ] ℝ) (hell : ‖ell‖ = 1) |
| {η : ℝ} (hη : 0 < η) : ∃ y : X, ‖y‖ ≤ 1 ∧ 1 - η < ell y := by |
| obtain ⟨u, hu, hval⟩ := ell.exists_lt_apply_of_lt_opNorm |
| (show 1 - η < ‖ell‖ by rw [hell]; linarith) |
| rw [Real.norm_eq_abs] at hval |
| by_cases hs : 0 ≤ ell u |
| · exact ⟨u, hu.le, by rwa [abs_of_nonneg hs] at hval⟩ |
| · refine ⟨-u, by simpa using hu.le, ?_⟩ |
| rw [abs_of_neg (lt_of_not_ge hs)] at hval |
| simpa only [map_neg] using hval |
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| / |
| theorem unitSlice_daugavet [Nontrivial X] (hsp : UnitSliceProperty X) : Daugavet X := by |
| intro ell v |
| by_cases he0 : ell = 0 |
| · simp [he0, ContinuousLinearMap.norm_id] |
| by_cases hv0 : v = 0 |
| · simp [hv0, ContinuousLinearMap.norm_id] |
| have he : 0 < ‖ell‖ := norm_pos_iff.mpr he0 |
| have hv : 0 < ‖v‖ := norm_pos_iff.mpr hv0 |
| let f : X →L[ℝ] ℝ := ‖ell‖⁻¹ • ell |
| have hf : ‖f‖ = 1 := norm_smul_inv_norm he0 |
| let w := ‖v‖⁻¹ • v |
| have hw : ‖w‖ = 1 := norm_smul_inv_norm hv0 |
| let a := ‖ell‖ * ‖v‖ |
| have ha : 0 < a := mul_pos he hv |
| let T : X →L[ℝ] X := ContinuousLinearMap.id ℝ X + ell.smulRight v |
| have hnrank : ‖ell.smulRight v‖ = a := ContinuousLinearMap.norm_smulRight_apply _ _ |
| apply le_antisymm |
| · have hb := norm_add_le (ContinuousLinearMap.id ℝ X) (ell.smulRight v) |
| simpa only [ContinuousLinearMap.norm_id] using hb |
| · rw [hnrank] |
| change 1 + a ≤ ‖T‖ |
| apply le_of_forall_pos_le_add |
| intro δ hδ |
| let η := min (1 / 2 : ℝ) (δ / (4 * (a + 1))) |
| have hη : 0 < η := by dsimp [η]; positivity |
| have hη1 : η < 1 := (min_le_left _ _).trans_lt (by norm_num) |
| have hηδ : η * (4 * (a + 1)) ≤ δ := |
| (le_div_iff₀ (by positivity)).mp (min_le_right _ _) |
| obtain ⟨y, hy, hfy, hfar⟩ := hsp w hw f (1 - η) |
| (normalized_functional_slice_nonempty f hf hη) η hη |
| obtain ⟨φ, hφ, hφsum⟩ := exists_dual_vector'' ℝ (w + y) |
| change φ (w + y) = ‖w + y‖ at hφsum |
| rw [map_add] at hφsum |
| have hφw : φ w ≤ 1 := by simpa only [hw] using dual_apply_le_norm φ hφ w |
| have hφy : φ y ≤ 1 := by simpa only [hy] using dual_apply_le_norm φ hφ y |
| have hwlo : 1 - η ≤ φ w := by linarith |
| have hylo : 1 - η ≤ φ y := by linarith |
| have hfylo : 1 - η ≤ f y := hfy.le |
| have hprod : (1 - η) ^ 2 ≤ f y * φ w := by |
| have hp := mul_le_mul hfylo hwlo (by linarith : 0 ≤ 1 - η) (by linarith : 0 ≤ f y) |
| nlinarith |
| have hmul := mul_le_mul_of_nonneg_left hprod ha.le |
| have heval : ell y = ‖ell‖ * f y := by |
| change ell y = ‖ell‖ * (‖ell‖⁻¹ * ell y) |
| rw [← mul_assoc, mul_inv_cancel₀ he.ne', one_mul] |
| have hvid : v = ‖v‖ • w := by |
| dsimp [w] |
| rw [smul_smul, mul_inv_cancel₀ hv.ne', one_smul] |
| have hφv : φ v = ‖v‖ * φ w := by conv_lhs => rw [hvid]; rw [map_smul, smul_eq_mul] |
| have hTφ : φ (T y) = φ y + a * (f y * φ w) := by |
| change φ (y + ell y • v) = _ |
| rw [map_add, map_smul, smul_eq_mul, heval, hφv] |
| dsimp [a] |
| ring |
| have hbound := dual_apply_le_norm φ hφ (T y) |
| have hop := T.le_opNorm y |
| rw [hy, mul_one] at hop |
| rw [hTφ] at hbound |
| nlinarith [mul_nonneg ha.le (sq_nonneg η)] |
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| end ExactModuli |
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| end |
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| end OAI |
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