| import OAI.Analysis.HyperbolicCones.GramProjection |
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| namespace OAI |
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| noncomputable section |
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| open scoped ContDiff RealInnerProductSpace |
| open Set ContinuousLinearMap |
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| universe u v |
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| namespace Paper256 |
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| theorem exists_smooth_kernel_extension |
| {W : Type u} {F : Type v} [NormedAddCommGroup W] [NormedSpace ℝ W] |
| [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] |
| (S : Set W) (hne : S.Nonempty) (A : W → F →L[ℝ] F) |
| (hA : ContDiff ℝ ∞ A) (hs : ∀ u ∈ S, IsSelfAdjoint (A u)) : |
| ∃ v ∈ S, ∃ (U : Set W) (P : W → F →L[ℝ] F), |
| IsOpen U ∧ v ∈ U ∧ ContDiffOn ℝ ∞ P U ∧ |
| ∀ u ∈ U, u ∈ S → P u = (A u).ker.starProjection := by |
| let ranks : Set ℕ := (fun u => Module.finrank ℝ (A u).range) '' S |
| have hfin : ranks.Finite := Set.finite_iff_bddAbove.mpr ⟨Module.finrank ℝ F, by |
| rintro _ ⟨u, _, rfl⟩ |
| exact Submodule.finrank_le _⟩ |
| obtain ⟨r, hr, hmax⟩ := Set.exists_max_image ranks id hfin (hne.image _) |
| obtain ⟨v, hv, rfl⟩ := hr |
| let R : Submodule ℝ F := (A v).range |
| let T : W → R →L[ℝ] F := fun u => (A u).comp R.subtypeL |
| have hTv : Function.Injective (T v) := by |
| apply LinearMap.ker_eq_bot.mp |
| apply LinearMap.ker_eq_bot'.mpr |
| intro x hx |
| change A v (x : F) = 0 at hx |
| have hxo : (x : F) ∈ Rᗮ := by |
| change (x : F) ∈ (A v).rangeᗮ |
| rw [(A v).orthogonal_range, (hs v hv).adjoint_eq] |
| exact hx |
| apply Subtype.ext |
| exact inner_self_eq_zero.mp (hxo x x.property) |
| have ht : ContDiff ℝ ∞ T := hA.clm_comp contDiff_const |
| have hadj : ContDiff ℝ ∞ (fun B : R →L[ℝ] F => B.adjoint) := |
| (ContinuousLinearMap.adjoint : (R →L[ℝ] F) ≃ₗᵢ[ℝ] (F →L[ℝ] R)).contDiff |
| let G : W → R →L[ℝ] R := fun u => (T u).adjoint.comp (T u) |
| have hg : ContDiff ℝ ∞ G := (hadj.comp ht).clm_comp ht |
| let U : Set W := {u | IsUnit (G u)} |
| have huopen : IsOpen U := by |
| change IsOpen (G ⁻¹' {K : R →L[ℝ] R | IsUnit K}) |
| apply IsOpen.preimage hg.continuous |
| exact Units.isOpen (R := R →L[ℝ] R) |
| have hvu : v ∈ U := gram_isUnit_of_injective (T v) hTv |
| refine ⟨v, hv, U, fun u => 1 - gramProjection (T u), huopen, hvu, ?_, ?_⟩ |
| · exact contDiffOn_const.sub (contDiffOn_gramProjection T U ht.contDiffOn (fun _ hu => hu)) |
| · intro u hu hus |
| have hTu : Function.Injective (T u) := by |
| have hGi := (ContinuousLinearMap.isUnit_iff_bijective.mp hu).1 |
| intro x y hxy |
| apply hGi |
| exact congrArg (T u).adjoint hxy |
| have hrange : (T u).range = (A u).range := by |
| apply Submodule.eq_of_le_of_finrank_le |
| · rintro _ ⟨x, rfl⟩ |
| exact ⟨(x : F), rfl⟩ |
| · rw [LinearMap.finrank_range_of_inj hTu] |
| exact hmax _ ⟨u, hus, rfl⟩ |
| exact one_sub_gramProjection_eq_kernel (T u) hu (A u) (hs u hus) hrange |
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| end Paper256 |
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| end |
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| end OAI |
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