| import OAI.Geometry.ComplexBundle.LineStage |
| import Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| /-! Continuous orthonormal frames and orthogonal complements. -/ |
|
|
| section |
| open Bundle Topology |
| open scoped BigOperators |
| namespace Kaplansky |
|
|
| section |
| open ComplexGrassmann |
| variable {ι : Type*} {K : ι → ℕ} {d : ℕ} |
|
|
| structure ContinuousIndependentStageFrame (a : LineStage K) (d : ℕ) where |
| vector : ∀ y, Fin d → a.Fiber y |
| continuous_vector : ∀ i, Continuous (fun y => TotalSpace.mk' a.Model y (vector y i)) |
| independent : ∀ y, LinearIndependent ℂ (vector y) |
|
|
| structure ContinuousOrthonormalStageFrame (a : LineStage K) (d : ℕ) where |
| vector : ∀ y, Fin d → a.Fiber y |
| continuous_vector : ∀ i, Continuous (fun y => TotalSpace.mk' a.Model y (vector y i)) |
| orthonormal : ∀ y, Orthonormal ℂ (vector y) |
|
|
| namespace ContinuousIndependentStageFrame |
| variable {a : LineStage K} |
|
|
| theorem continuous_gram (f : ContinuousIndependentStageFrame a d) : |
| Continuous (fun y => Matrix.gram ℂ (f.vector y)) := by |
| apply continuous_pi |
| intro i |
| apply continuous_pi |
| intro j |
| exact continuous_lineStage_inner_over a id continuous_id (fun y => f.vector y i) |
| (fun y => f.vector y j) (f.continuous_vector i) (f.continuous_vector j) |
|
|
| def normalize (f : ContinuousIndependentStageFrame a d) : ContinuousOrthonormalStageFrame a d where |
| vector y := metricFrame (f.vector y) |
| orthonormal y := metricFrame_orthonormal _ (f.independent y) |
| continuous_vector j := by |
| have hc := continuous_metricFrame_coefficients _ f.continuous_gram |
| (fun y => Matrix.posDef_gram_of_linearIndependent (f.independent y)) |
| apply continuousOn_univ.mp |
| unfold metricFrame frameMul |
| apply continuousOn_bundle_sum (𝕜 := ℂ) |
| intro i _ |
| apply continuousOn_bundle_smul |
| · exact ((continuous_apply j).comp ((continuous_apply i).comp hc)).continuousOn |
| · exact (f.continuous_vector i).continuousOn |
|
|
| theorem normalize_span (f : ContinuousIndependentStageFrame a d) (y : ProjectiveProduct K) : |
| Submodule.span ℂ (Set.range (f.normalize.vector y)) = |
| Submodule.span ℂ (Set.range (f.vector y)) := |
| span_metricFrame _ (f.independent y) |
|
|
| end ContinuousIndependentStageFrame |
| end |
|
|
| open ComplexGrassmann |
| variable {ι : Type*} {K : ι → ℕ} {d : ℕ} {a : LineStage K} |
| namespace ContinuousOrthonormalStageFrame |
|
|
| variable (f : ContinuousOrthonormalStageFrame a d) |
|
|
| abbrev ImageTotal := {z : TotalSpace a.Model a.Fiber |
| z.2 ∈ Submodule.span ℂ (Set.range (f.vector z.proj))} |
|
|
| theorem continuous_synthesis : |
| Continuous (fun z : ProjectiveProduct K × EuclideanSpace ℂ (Fin d) => |
| TotalSpace.mk' a.Model z.1 (frameSynthesis (f.vector z.1) z.2)) := by |
| apply continuousOn_univ.mp |
| change ContinuousOn (fun z : ProjectiveProduct K × EuclideanSpace ℂ (Fin d) => |
| TotalSpace.mk' a.Model z.1 (∑ i, z.2 i • f.vector z.1 i)) Set.univ |
| apply continuousOn_bundle_sum_over (𝕜 := ℂ) _ continuous_fst.continuousOn |
| intro i _ |
| apply continuousOn_bundle_smul_over _ continuous_fst.continuousOn |
| · exact ((continuous_apply i).comp ((EuclideanSpace.equiv _ ℂ).continuous.comp continuous_snd)).continuousOn |
| · exact ((f.continuous_vector i).comp continuous_fst).continuousOn |
|
|
| theorem continuous_analysis : |
| Continuous (fun z : TotalSpace a.Model a.Fiber => frameAnalysis (f.vector z.proj) z.2) := by |
| have hc : Continuous (fun z : TotalSpace a.Model a.Fiber => fun i : Fin d => |
| inner ℂ (f.vector z.proj i) z.2) := by |
| apply continuous_pi |
| intro i |
| exact continuous_lineStage_inner_over a _ (FiberBundle.continuous_proj _ _) |
| (fun z => f.vector z.proj i) (fun z => z.2) |
| ((f.continuous_vector i).comp (FiberBundle.continuous_proj _ _)) continuous_id |
| exact (EuclideanSpace.equiv (Fin d) ℂ).symm.continuous.comp hc |
|
|
| /-- A homeomorphism with the subspace topology inherited from the actual |
| ambient bundle, not a separately declared topology on an abstract image. -/ |
| def imageTotalHomeomorph : ProjectiveProduct K × EuclideanSpace ℂ (Fin d) ≃ₜ f.ImageTotal where |
| toFun z := ⟨⟨z.1,frameSynthesis (f.vector z.1) z.2⟩,frameSynthesis_mem_span _ _⟩ |
| invFun z := (z.val.proj,frameAnalysis (f.vector z.val.proj) z.val.2) |
| left_inv z := Prod.ext rfl (frameAnalysis_synthesis _ (f.orthonormal z.1) z.2) |
| right_inv z := by |
| apply Subtype.ext |
| obtain ⟨⟨y,v⟩,hv⟩ := z |
| exact congrArg (TotalSpace.mk' a.Model y) (frameSynthesis_analysis _ (f.orthonormal y) hv) |
| continuous_toFun := f.continuous_synthesis.subtype_mk _ |
| continuous_invFun := |
| ((FiberBundle.continuous_proj _ _).comp continuous_subtype_val).prodMk |
| (f.continuous_analysis.comp continuous_subtype_val) |
| |
| @[simp] theorem imageTotalHomeomorph_base |
| (z : ProjectiveProduct K × EuclideanSpace ℂ (Fin d)) : |
| (f.imageTotalHomeomorph z).val.proj=z.1 := rfl |
| |
| theorem imageTotalHomeomorph_fiber_linear (y : ProjectiveProduct K) : |
| ∃ L : EuclideanSpace ℂ (Fin d) →ₗ[ℂ] a.Fiber y, |
| ∀ v, (f.imageTotalHomeomorph (y,v)).val.2=L v := |
| ⟨frameSynthesis (f.vector y),fun _ => rfl⟩ |
| |
| def projection (y : ProjectiveProduct K) : a.Fiber y →ₗ[ℂ] a.Fiber y := |
| (frameSynthesis (f.vector y)).comp (frameAnalysis (f.vector y)) |
| |
| theorem projection_idempotent (y : ProjectiveProduct K) (v : a.Fiber y) : |
| f.projection y (f.projection y v)=f.projection y v := by |
| exact congrArg (frameSynthesis (f.vector y)) |
| (frameAnalysis_synthesis _ (f.orthonormal y) (frameAnalysis (f.vector y) v)) |
| |
| theorem projection_mem (y : ProjectiveProduct K) (v : a.Fiber y) : |
| f.projection y v ∈ Submodule.span ℂ (Set.range (f.vector y)) := frameSynthesis_mem_span _ _ |
| |
| theorem projection_eq_self (y : ProjectiveProduct K) (v : a.Fiber y) : |
| f.projection y v = v ↔ v ∈ Submodule.span ℂ (Set.range (f.vector y)) := by |
| constructor |
| · intro h |
| rw [← h] |
| exact f.projection_mem y v |
| · exact frameSynthesis_analysis _ (f.orthonormal y) |
| |
| theorem continuous_projection : |
| Continuous (fun z : TotalSpace a.Model a.Fiber => |
| TotalSpace.mk' a.Model z.proj (f.projection z.proj z.2)) := |
| f.continuous_synthesis.comp ((FiberBundle.continuous_proj _ _).prodMk f.continuous_analysis) |
|
|
| theorem image_finrank (y : ProjectiveProduct K) : |
| Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))=d := |
| frameSpan_finrank _ (f.orthonormal y) |
|
|
| end ContinuousOrthonormalStageFrame |
| end Kaplansky |
|
|
| end |
|
|
| section |
| open scoped BigOperators |
| namespace Kaplansky |
| variable {d : ℕ} {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
|
|
| theorem frameSynthesis_inner (v : Fin d → E) (hv : Orthonormal ℂ v) |
| (x y : EuclideanSpace ℂ (Fin d)) : |
| inner ℂ (frameSynthesis v x) (frameSynthesis v y)=inner ℂ x y := by |
| have h := hv.inner_sum (fun i => x i) (fun i => y i) Finset.univ |
| change inner ℂ (∑ i, x i • v i) (∑ i, y i • v i) = _ |
| rw [h] |
| change (∑ i, star (x i)*y i) = ∑ i, inner ℂ (x i) (y i) |
| simp only [RCLike.inner_apply,starRingEnd_apply,mul_comm] |
|
|
| def frameSynthesisIsometry (v : Fin d → E) (hv : Orthonormal ℂ v) : |
| EuclideanSpace ℂ (Fin d) →ₗᵢ[ℂ] E where |
| toLinearMap := frameSynthesis v |
| norm_map' := (LinearMap.norm_map_iff_inner_map_map _).mpr (frameSynthesis_inner v hv) |
| |
| open ComplexGrassmann |
| namespace ContinuousOrthonormalStageFrame |
| variable {ι : Type*} {K : ι → ℕ} {a : LineStage K} |
| |
| theorem projection_error_inner (f : ContinuousOrthonormalStageFrame a d) |
| (y : ProjectiveProduct K) (v : a.Fiber y) (i : Fin d) : |
| inner ℂ (f.vector y i) (v-f.projection y v)=0 := by |
| have hi := congrArg (fun x : EuclideanSpace ℂ (Fin d) => x i) |
| (frameAnalysis_synthesis (f.vector y) (f.orthonormal y) (frameAnalysis (f.vector y) v)) |
| change inner ℂ (f.vector y i) (f.projection y v)=inner ℂ (f.vector y i) v at hi |
| exact (inner_sub_right (𝕜 := ℂ) (f.vector y i) v (f.projection y v)).trans |
| (sub_eq_zero.mpr hi.symm) |
| |
| end ContinuousOrthonormalStageFrame |
| end Kaplansky |
| |
| end |
| |
| open Bundle Topology |
| open scoped BigOperators |
| namespace Kaplansky |
| open ComplexGrassmann |
| namespace ContinuousOrthonormalStageFrame |
| variable {ι : Type*} {K : ι → ℕ} {a : LineStage K} {d : ℕ} |
| variable (f : ContinuousOrthonormalStageFrame a d) |
| |
| def complementProjection (y : ProjectiveProduct K) : a.Fiber y →ₗ[ℂ] a.Fiber y := |
| LinearMap.id - f.projection y |
| |
| theorem complementProjection_mem (y : ProjectiveProduct K) (v : a.Fiber y) : |
| f.complementProjection y v ∈ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ := by |
| apply (Submodule.mem_orthogonal _ _).mpr |
| intro u hu |
| induction hu using Submodule.span_induction with |
| | mem u hu => |
| obtain ⟨i, rfl⟩ := hu |
| exact f.projection_error_inner y v i |
| | zero => exact inner_zero_left _ |
| | add u w hu hw ihu ihw => |
| exact (inner_add_left u w _).trans (by rw [ihu, ihw, zero_add]) |
| | smul c u hu ihu => |
| exact (inner_smul_left u _ c).trans (by rw [ihu, mul_zero]) |
| |
| theorem projection_eq_zero_of_orthogonal (y : ProjectiveProduct K) (v : a.Fiber y) |
| (hv : v ∈ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ) : f.projection y v=0 := by |
| have ha : frameAnalysis (f.vector y) v=0 := by |
| apply (EuclideanSpace.equiv (Fin d) ℂ).injective |
| funext i |
| exact (Submodule.mem_orthogonal _ _).mp hv _ (Submodule.subset_span ⟨i,rfl⟩) |
| change frameSynthesis (f.vector y) (frameAnalysis (f.vector y) v)=0 |
| exact (congrArg (frameSynthesis (f.vector y)) ha).trans |
| (frameSynthesis (f.vector y)).map_zero |
| |
| theorem complementProjection_range (y : ProjectiveProduct K) : |
| LinearMap.range (f.complementProjection y) = (Submodule.span ℂ (Set.range (f.vector y)))ᗮ := by |
| apply le_antisymm |
| · rintro _ ⟨v,rfl⟩ |
| exact f.complementProjection_mem y v |
| · intro v hv |
| refine ⟨v,?_⟩ |
| change v-f.projection y v=v |
| rw [f.projection_eq_zero_of_orthogonal y v hv,sub_zero] |
| |
| theorem continuous_complementProjection : |
| Continuous (fun z : TotalSpace a.Model a.Fiber => |
| TotalSpace.mk' a.Model z.proj (f.complementProjection z.proj z.2)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro z |
| rw [FiberBundle.continuousAt_totalSpace] |
| refine ⟨(FiberBundle.continuous_proj _ _).continuousAt,?_⟩ |
| let e := trivializationAt a.Model a.Fiber z.proj |
| have hc := ((FiberBundle.continuousAt_totalSpace a.Model _ (x₀ := z)).mp (continuous_id (X := TotalSpace a.Model a.Fiber)).continuousAt).2 |
| have hp := ((FiberBundle.continuousAt_totalSpace a.Model _ (x₀ := z)).mp f.continuous_projection.continuousAt).2 |
| apply (hc.sub hp).congr_of_eventuallyEq |
| have hn := e.open_baseSet.mem_nhds (mem_baseSet_trivializationAt a.Model a.Fiber z.proj) |
| filter_upwards [(FiberBundle.continuous_proj a.Model a.Fiber).continuousAt hn] with w hw |
| exact (e.linear ℂ hw).map_sub w.2 (f.projection w.proj w.2) |
|
|
| theorem complement_finrank (y : ProjectiveProduct K) : |
| Module.finrank ℂ (Submodule.span ℂ (Set.range (f.vector y)))ᗮ + d = Fintype.card a.Slot := by |
| let V : Fin d → EuclideanSpace ℂ a.Slot := f.vector y |
| have hV : Orthonormal ℂ V := f.orthonormal y |
| change Module.finrank ℂ (Submodule.span ℂ (Set.range V))ᗮ + d = _ |
| have h := (Submodule.span ℂ (Set.range V)).finrank_add_finrank_orthogonal |
| rw [frameSpan_finrank V hV, finrank_euclideanSpace] at h |
| omega |
|
|
| end ContinuousOrthonormalStageFrame |
| end Kaplansky |
|
|
| end |
|
|
| end OAI |
|
|
