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2fca478 | 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 | 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
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