File size: 2,890 Bytes
9eeb9c7 | 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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Analysis.HyperbolicCones.GramProjection
namespace OAI
noncomputable section
open scoped ContDiff RealInnerProductSpace
open Set ContinuousLinearMap
universe u v
namespace Paper256
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
end Paper256
end
end OAI
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