| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped InnerProductSpace | |
| namespace RPISingularHarnack | |
| theorem projection_eq_of_pairing_eq {H : Type*} [NormedAddCommGroup H] | |
| [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection] | |
| {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) : | |
| K.orthogonalProjectionOnto u=K.orthogonalProjectionOnto e := by | |
| apply ext_inner_right ℝ | |
| intro w | |
| simp only [K.inner_orthogonalProjectionOnto_eq_of_mem_right] | |
| exact h w w.property | |
| theorem harmonic_projection_error_le {H : Type*} [NormedAddCommGroup H] | |
| [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection] | |
| {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) : | |
| ‖u-(u-K.starProjection u)‖≤‖e‖ := by | |
| rw [sub_sub_cancel,Submodule.starProjection_apply, | |
| projection_eq_of_pairing_eq K h,Submodule.norm_coe] | |
| exact K.norm_orthogonalProjectionOnto_apply_le e | |
| theorem harmonic_projection_exists {H : Type*} [NormedAddCommGroup H] | |
| [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection] | |
| {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) : | |
| ∃ g : H, g∈Kᗮ ∧ u-g∈K ∧ ‖u-g‖≤‖e‖ := by | |
| refine ⟨u-K.starProjection u,K.sub_starProjection_mem_orthogonal u,?_, | |
| harmonic_projection_error_le K h⟩ | |
| rw [sub_sub_cancel] | |
| exact (K.orthogonalProjectionOnto u).property | |
| end RPISingularHarnack | |
| end | |
| end OAI | |