File size: 6,474 Bytes
59d1c80 | 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.Geometry.IsometricImmersion.Assembly.SupportedPerturbations
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Sequences
import Mathlib.Topology.Baire.CompleteMetrizable
import Mathlib.Topology.Baire.Lemmas
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise Distributions
open Set Filter TopologicalSpace
open scoped BoundedContinuousFunction
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V]
abbrev SupportedJetProduct (V : Type*) [NormedAddCommGroup V] [NormedSpace ℝ V] :=
Π k : ℕ, Coord →ᵇ (Coord [×k]→L[ℝ] V)
def supportedJetMap (K : Compacts Coord) :
ContDiffMapSupportedIn Coord V ⊤ K →L[ℝ] SupportedJetProduct V :=
ContinuousLinearMap.pi (ContDiffMapSupportedIn.structureMapCLM ℝ ⊤)
omit [CompleteSpace V] in
theorem supportedJetMap_isUniformEmbedding (K : Compacts Coord) :
IsUniformEmbedding (supportedJetMap K :
ContDiffMapSupportedIn Coord V ⊤ K → SupportedJetProduct V) :=
ContDiffMapSupportedIn.isUniformEmbedding_pi_structureMapCLM (𝕜 := ℝ) (n := ⊤)
omit [CompleteSpace V] in
theorem supportedJetMap_isClosed_range (K : Compacts Coord) :
IsClosed (Set.range (supportedJetMap K :
ContDiffMapSupportedIn Coord V ⊤ K → SupportedJetProduct V)) := by
apply isSeqClosed_iff_isClosed.mp
intro u v hu ht
choose f hf using hu
have hfconv : Tendsto (fun n => supportedJetMap K (f n)) atTop (𝓝 v) := by
simpa only [hf] using ht
have hconv (k : ℕ) : TendstoUniformly
(fun n => iteratedFDeriv ℝ k (f n)) (v k) atTop := by
have hcoord := ((continuous_apply k).tendsto v).comp hfconv
change Tendsto (fun n => ContDiffMapSupportedIn.structureMapCLM ℝ ⊤ k (f n))
atTop (𝓝 (v k)) at hcoord
have h := BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hcoord
simpa only [ContDiffMapSupportedIn.structureMapCLM_top_apply] using h
let C (k : ℕ) :=
continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (k + 1) => Coord) V
have hderivative (k : ℕ) (x : Coord) :
HasFDerivAt (fun y => v k y) (C k (v (k + 1) x)) x := by
have hnext : TendstoUniformly
(fun n => C k ∘ iteratedFDeriv ℝ (k + 1) (f n)) (C k ∘ v (k + 1)) atTop :=
(C k).isometry.uniformContinuous.comp_tendstoUniformly (hconv (k + 1))
have hfn (n : ℕ) (y : Coord) : HasFDerivAt (iteratedFDeriv ℝ k (f n))
(C k (iteratedFDeriv ℝ (k + 1) (f n) y)) y := by
have hs : ContDiff ℝ ∞ (f n) := (f n).contDiff
have hd := ((hs.contDiffAt (x := y)).differentiableAt_iteratedFDeriv
(ENat.natCast_lt_of_coe_top_le_withTop le_rfl k)).hasFDerivAt
simpa only [fderiv_iteratedFDeriv, Function.comp_apply, C] using hd
exact hasFDerivAt_of_tendstoUniformly hnext hfn (fun y => (hconv k).tendsto_at y) x
let C0 := continuousMultilinearCurryFin0 ℝ Coord V
let g : Coord → V := fun x => C0 (v 0 x)
have hjet : ∀ k : ℕ, iteratedFDeriv ℝ k g = (v k : Coord → Coord [×k]→L[ℝ] V) := by
intro k
induction k with
| zero =>
funext x
change C0.symm (C0 (v 0 x)) = v 0 x
exact C0.symm_apply_apply _
| succ k ih =>
rw [iteratedFDeriv_succ_eq_comp_left, ih]
funext x
change (C k).symm (fderiv ℝ (fun y => v k y) x) = v (k + 1) x
rw [(hderivative k x).fderiv]
exact (C k).symm_apply_apply _
have hg : ContDiff ℝ ∞ g := by
apply contDiff_iff_continuous_differentiable.mpr
constructor
· intro k _
rw [hjet k]
exact (v k).continuous
· intro k _
rw [hjet k]
exact fun x => (hderivative k x).differentiableAt
have hzero : EqOn g 0 (K : Set Coord)ᶜ := by
intro x hx
have hlim := (hconv 0).tendsto_at x
have hconst : Tendsto (fun n => iteratedFDeriv ℝ 0 (f n) x) atTop (𝓝 0) := by
have hz (n : ℕ) : iteratedFDeriv ℝ 0 (f n) x = 0 :=
(f n).iteratedFDeriv_zero_on_compl hx
simpa only [hz] using (tendsto_const_nhds (x := (0 : Coord [×0]→L[ℝ] V)))
have hvzero : v 0 x = 0 := tendsto_nhds_unique hlim hconst
change C0 (v 0 x) = 0
rw [hvzero, map_zero]
let φ : ContDiffMapSupportedIn Coord V ⊤ K := ⟨g, hg, hzero⟩
refine ⟨φ, ?_⟩
funext k
apply BoundedContinuousFunction.ext
intro x
change ContDiffMapSupportedIn.structureMapCLM ℝ ⊤ k φ x = v k x
rw [ContDiffMapSupportedIn.structureMapCLM_top_apply]
exact congrFun (hjet k) x
theorem supportedSmooth_completeSpace (K : Compacts Coord) :
CompleteSpace (ContDiffMapSupportedIn Coord V ⊤ K) :=
(supportedJetMap_isUniformEmbedding (V := V) K).isUniformInducing.completeSpace
(supportedJetMap_isClosed_range (V := V) K).isComplete
theorem supportedSmooth_completelyMetrizable (K : Compacts Coord) :
IsCompletelyMetrizableSpace (ContDiffMapSupportedIn Coord V ⊤ K) := by
have (k : ℕ) : IsCompletelyMetrizableSpace (Coord →ᵇ (Coord [×k]→L[ℝ] V)) := by
have hc : CompleteSpace (Coord →ᵇ (Coord [×k]→L[ℝ] V)) := inferInstance
refine ⟨⟨inferInstance, ?_, ?_⟩⟩
· rfl
· exact hc
have : IsCompletelyMetrizableSpace (SupportedJetProduct V) := inferInstance
have he : Topology.IsClosedEmbedding (supportedJetMap K :
ContDiffMapSupportedIn Coord V ⊤ K → SupportedJetProduct V) :=
⟨(supportedJetMap_isUniformEmbedding K).isEmbedding, supportedJetMap_isClosed_range K⟩
exact he.IsCompletelyMetrizableSpace
theorem symmetricPerturbation_completeSpace : CompleteSpace SymmetricPerturbation := by
let : CompleteSpace SupportedTensor := supportedSmooth_completeSpace patchCompact
let : IsClosed (symmetricSupportedSubmodule : Set SupportedTensor) :=
symmetricSupportedSubmodule_isClosed
infer_instance
theorem symmetricPerturbation_completelyMetrizable :
IsCompletelyMetrizableSpace SymmetricPerturbation := by
let : IsCompletelyMetrizableSpace SupportedTensor := supportedSmooth_completelyMetrizable patchCompact
exact symmetricSupportedSubmodule_isClosed.isClosedEmbedding_subtypeVal.IsCompletelyMetrizableSpace
theorem symmetricPerturbation_baire : BaireSpace SymmetricPerturbation := by
let : IsCompletelyMetrizableSpace SymmetricPerturbation := symmetricPerturbation_completelyMetrizable
infer_instance
end SmoothLocal.Perturbation
end
end OAI
|