openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /AffiliatedTransportSupplement.lean
| import Mathlib.Analysis.InnerProductSpace.LinearPMap | |
| import Mathlib.Analysis.InnerProductSpace.Projection.Basic | |
| import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.AffiliatedTransport | |
| open Filter Topology | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] | |
| def imageOn (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) : Submodule ℂ E := | |
| (p.comap H.domain.subtype).map H.toFun | |
| def closedImageOn (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) : Submodule ℂ E := | |
| (imageOn H p).topologicalClosure | |
| instance [CompleteSpace E] (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) : | |
| CompleteSpace (closedImageOn H p) := by | |
| unfold closedImageOn | |
| infer_instance | |
| @[simp] theorem mem_imageOn {H : E →ₗ.[ℂ] E} {p : Submodule ℂ E} {y : E} : | |
| y ∈ imageOn H p ↔ ∃ x : H.domain, (x : E) ∈ p ∧ H x = y := by | |
| rfl | |
| theorem imageOn_le_closedImageOn (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) : | |
| imageOn H p ≤ closedImageOn H p := | |
| Submodule.le_topologicalClosure _ | |
| theorem image_mem_closedImageOn (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (x : H.domain) (hx : (x : E) ∈ p) : H x ∈ closedImageOn H p := | |
| imageOn_le_closedImageOn H p ((mem_imageOn).2 ⟨x, hx, rfl⟩) | |
| theorem imageOn_invariant (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (A B : E →L[ℂ] E) | |
| (hD : ∀ x : H.domain, A (x : E) ∈ H.domain) | |
| (hI : ∀ x : H.domain, H ⟨A (x : E), hD x⟩ = B (H x)) | |
| (hp : ∀ x ∈ p, A x ∈ p) : | |
| ∀ y ∈ imageOn H p, B y ∈ imageOn H p := by | |
| rintro y ⟨x, hx, rfl⟩ | |
| exact (mem_imageOn).2 ⟨⟨A (x : E), hD x⟩, hp x hx, hI x⟩ | |
| theorem closedImageOn_invariant (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (A B : E →L[ℂ] E) | |
| (hD : ∀ x : H.domain, A (x : E) ∈ H.domain) | |
| (hI : ∀ x : H.domain, H ⟨A (x : E), hD x⟩ = B (H x)) | |
| (hp : ∀ x ∈ p, A x ∈ p) : | |
| ∀ y ∈ closedImageOn H p, B y ∈ closedImageOn H p := by | |
| have hmap : (imageOn H p).map B.toLinearMap ≤ imageOn H p := by | |
| rintro y ⟨x, hx, rfl⟩ | |
| exact imageOn_invariant H p A B hD hI hp x hx | |
| intro y hy | |
| change B y ∈ (imageOn H p).topologicalClosure | |
| apply Submodule.topologicalClosure_mono hmap | |
| apply Submodule.topologicalClosure_map B (imageOn H p) | |
| exact ⟨y, hy, rfl⟩ | |
| theorem closedImageOn_invariant_of_commuting (H : E →ₗ.[ℂ] E) | |
| (p : Submodule ℂ E) (W : E →L[ℂ] E) | |
| (hD : ∀ x : H.domain, W (x : E) ∈ H.domain) | |
| (hC : ∀ x : H.domain, H ⟨W (x : E), hD x⟩ = W (H x)) | |
| (hp : ∀ x ∈ p, W x ∈ p) : | |
| ∀ y ∈ closedImageOn H p, W y ∈ closedImageOn H p := | |
| closedImageOn_invariant H p W W hD hC hp | |
| theorem mem_orthogonal_closedImageOn (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (hH : H.IsFormalAdjoint H) (z : H.domain) | |
| (hz : H z ∈ pᗮ) : (z : E) ∈ (closedImageOn H p)ᗮ := by | |
| rw [closedImageOn, Submodule.orthogonal_closure] | |
| rw [Submodule.mem_orthogonal] | |
| rintro y ⟨x, hx, rfl⟩ | |
| change inner ℂ (H x) (z : E) = 0 | |
| exact (hH x z).trans (Submodule.inner_right_of_mem_orthogonal hx hz) | |
| theorem inverse_imageOn_le_orthogonal (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (hH : H.IsFormalAdjoint H) (hinj : H.toFun.ker = ⊥) : | |
| imageOn H.inverse pᗮ ≤ (closedImageOn H p)ᗮ := by | |
| rintro y ⟨η, hη, rfl⟩ | |
| have hηrange : (η : E) ∈ LinearMap.range H.toFun := by | |
| rw [← LinearPMap.inverse_domain] | |
| exact η.property | |
| obtain ⟨z, hz⟩ := hηrange | |
| have hker : H.ker = ⊥ := by simp [LinearPMap.ker, hinj] | |
| have heq : H.inverse η = (z : E) := LinearPMap.inverse_apply_eq hker hz | |
| change H.inverse η ∈ (closedImageOn H p)ᗮ | |
| rw [heq] | |
| apply mem_orthogonal_closedImageOn H p hH z | |
| change H.toFun z ∈ pᗮ | |
| rw [hz] | |
| exact hη | |
| theorem inverse_closedImageOn_le_orthogonal (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (hH : H.IsFormalAdjoint H) (hinj : H.toFun.ker = ⊥) : | |
| closedImageOn H.inverse pᗮ ≤ (closedImageOn H p)ᗮ := by | |
| exact (imageOn H.inverse pᗮ).topologicalClosure_minimal | |
| (inverse_imageOn_le_orthogonal H p hH hinj) | |
| (closedImageOn H p).isClosed_orthogonal | |
| theorem orthogonal_invariant_of_inverse (p : Submodule ℂ E) | |
| (W : E ≃ₗᵢ[ℂ] E) (hp : ∀ x ∈ p, W.symm x ∈ p) : | |
| ∀ y ∈ pᗮ, W y ∈ pᗮ := by | |
| intro y hy | |
| rw [Submodule.mem_orthogonal] | |
| intro x hx | |
| have h := W.inner_map_map (W.symm x) y | |
| rw [W.apply_symm_apply] at h | |
| exact h.trans (Submodule.inner_right_of_mem_orthogonal (hp x hx) hy) | |
| theorem starProjection_commutes_of_reducing (p : Submodule ℂ E) | |
| [p.HasOrthogonalProjection] (W : E ≃ₗᵢ[ℂ] E) | |
| (hp : ∀ x ∈ p, W x ∈ p) (hp' : ∀ x ∈ p, W.symm x ∈ p) : | |
| ∀ x, p.starProjection (W x) = W (p.starProjection x) := by | |
| intro x | |
| apply p.eq_starProjection_of_mem_orthogonal | |
| · exact hp _ (p.starProjection_apply_mem x) | |
| · have h := orthogonal_invariant_of_inverse p W hp' _ | |
| (p.sub_starProjection_mem_orthogonal x) | |
| simpa only [map_sub] using h | |
| theorem closedImageOn_map_eq (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) | |
| (W : E ≃L[ℂ] E) | |
| (hD : ∀ x : H.domain, W (x : E) ∈ H.domain) | |
| (hD' : ∀ x : H.domain, W.symm (x : E) ∈ H.domain) | |
| (hC : ∀ x : H.domain, H ⟨W (x : E), hD x⟩ = W (H x)) | |
| (hC' : ∀ x : H.domain, H ⟨W.symm (x : E), hD' x⟩ = W.symm (H x)) | |
| (hp : ∀ x ∈ p, W x ∈ p) (hp' : ∀ x ∈ p, W.symm x ∈ p) : | |
| (closedImageOn H p).map W.toLinearMap = closedImageOn H p := by | |
| apply le_antisymm | |
| · rintro y ⟨x, hx, rfl⟩ | |
| exact closedImageOn_invariant_of_commuting H p W.toContinuousLinearMap hD hC hp x hx | |
| · intro y hy | |
| refine ⟨W.symm y, ?_, W.apply_symm_apply y⟩ | |
| exact closedImageOn_invariant_of_commuting H p W.symm.toContinuousLinearMap | |
| hD' hC' hp' y hy | |
| theorem closedImageOn_starProjection_commutes [CompleteSpace E] | |
| (H : E →ₗ.[ℂ] E) (p : Submodule ℂ E) (W : E ≃ₗᵢ[ℂ] E) | |
| (hD : ∀ x : H.domain, W (x : E) ∈ H.domain) | |
| (hD' : ∀ x : H.domain, W.symm (x : E) ∈ H.domain) | |
| (hC : ∀ x : H.domain, H ⟨W (x : E), hD x⟩ = W (H x)) | |
| (hC' : ∀ x : H.domain, H ⟨W.symm (x : E), hD' x⟩ = W.symm (H x)) | |
| (hp : ∀ x ∈ p, W x ∈ p) (hp' : ∀ x ∈ p, W.symm x ∈ p) : | |
| ∀ x, (closedImageOn H p).starProjection (W x) = | |
| W ((closedImageOn H p).starProjection x) := by | |
| apply starProjection_commutes_of_reducing | |
| · exact closedImageOn_invariant_of_commuting H p W.toLinearIsometry.toContinuousLinearMap hD hC hp | |
| · exact closedImageOn_invariant_of_commuting H p W.symm.toLinearIsometry.toContinuousLinearMap hD' hC' hp' | |
| theorem trace_eq_of_cutoff_estimates {t p : ℝ} {e : ℕ → ℝ} | |
| (he : Tendsto e atTop (𝓝 1)) | |
| (hlower : ∀ n, p + e n - 1 ≤ t) | |
| (hupper : ∀ n, (1 - p) + e n - 1 ≤ 1 - t) : t = p := by | |
| have h₁ : p ≤ t := by | |
| have hlim : Tendsto (fun n => p + e n - 1) atTop (𝓝 p) := by | |
| convert (tendsto_const_nhds.add he).sub tendsto_const_nhds using 1; simp | |
| exact le_of_tendsto hlim (Filter.Eventually.of_forall hlower) | |
| have h₂ : 1 - p ≤ 1 - t := by | |
| have hlim : Tendsto (fun n => (1 - p) + e n - 1) atTop (𝓝 (1 - p)) := by | |
| convert (tendsto_const_nhds.add he).sub tendsto_const_nhds using 1; simp | |
| exact le_of_tendsto hlim (Filter.Eventually.of_forall hupper) | |
| linarith | |
| end Invariant233.AffiliatedTransport | |
| end | |
| end OAI | |