| import Mathlib |
| import OAI.Geometry.DoublingHilbert.CoordinateSupport |
| import OAI.Geometry.DoublingHilbert.Main |
|
|
| namespace OAI |
|
|
| open Set Metric |
| open scoped BigOperators |
| noncomputable section |
| namespace CompactBanach |
| open DoublingHilbert |
|
|
| noncomputable def coordinateMap {B : Type*} [NormedAddCommGroup B] [NormedSpace ℝ B] |
| (I : Finset ℕ) (v : I → B) : RealL2 →ₗ[ℝ] B where |
| toFun z := ∑ i : I, z i • v i |
| map_add' z w := by simp [add_smul, Finset.sum_add_distrib] |
| map_smul' c z := by simp [Finset.smul_sum, smul_smul] |
|
|
| theorem coordinateMap_single {B : Type*} [NormedAddCommGroup B] [NormedSpace ℝ B] |
| (I : Finset ℕ) (v : I → B) (hv : ∀ i, ‖v i‖ ≤ 1) (j : ℕ) (a : ℝ) : |
| ‖coordinateMap I v (lp.single 2 j a)‖ ≤ |a| := by |
| classical |
| change ‖∑ i : I, (lp.single 2 j a : RealL2) i • v i‖ ≤ |a| |
| by_cases hj : j ∈ I |
| · have he : (∑ i : I, (lp.single 2 j a : RealL2) i • v i) = a • v ⟨j, hj⟩ := by |
| rw [Finset.sum_eq_single ⟨j, hj⟩] |
| · simp [lp.single_apply] |
| · intro i _ hi |
| have hne : (i : ℕ) ≠ j := by |
| intro heq |
| exact hi (Subtype.ext heq) |
| rw [lp.single_apply_ne (E := fun _ : ℕ => ℝ) 2 j a hne, zero_smul] |
| · simp |
| rw [he, norm_smul, Real.norm_eq_abs] |
| simpa using mul_le_mul_of_nonneg_left (hv ⟨j, hj⟩) (abs_nonneg a) |
| · have he : (∑ i : I, (lp.single 2 j a : RealL2) i • v i) = 0 := by |
| apply Finset.sum_eq_zero |
| intro i _ |
| have hne : (i : ℕ) ≠ j := by rintro rfl |
| rw [lp.single_apply_ne (E := fun _ : ℕ => ℝ) 2 j a hne, zero_smul] |
| rw [he, norm_zero] |
| exact abs_nonneg a |
|
|
| theorem coordinateMap_lower {B : Type*} [NormedAddCommGroup B] [NormedSpace ℝ B] |
| (I : Finset ℕ) (v : I → B) |
| (hv : ∀ (c : I → ℝ) i, |c i| ≤ ‖∑ j, c j • v j‖) |
| (z : RealL2) (hz : ∀ i ∉ I, z i = 0) : |
| ∀ i, |z i| ≤ ‖coordinateMap I v z‖ := by |
| intro i |
| by_cases hi : i ∈ I |
| · exact hv (fun j => z j) ⟨i, hi⟩ |
| · rw [hz i hi, abs_zero] |
| exact norm_nonneg _ |
|
|
| end CompactBanach |
| end |
|
|
| end OAI |
|
|
