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110f4f3 | 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 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
namespace WeakMTWTransport
universe u w
variable {K : Type w} {E F : Type u} [TopologicalSpace K] [CompactSpace K]
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
local instance pathCalculusNormedAddCommGroupContinuousLinearMap :
NormedAddCommGroup (E →L[ℝ] F) := ContinuousLinearMap.toNormedAddCommGroup
local instance pathCalculusNormedSpaceContinuousLinearMap :
NormedSpace ℝ (E →L[ℝ] F) := ContinuousLinearMap.toNormedSpace
local instance pathCalculusNormedAddCommGroupContinuousMap :
NormedAddCommGroup C(K,E →L[ℝ] F) := ContinuousMap.instNormedAddCommGroup
def pathLinearApply (A : C(K,E →L[ℝ] F)) : C(K,E) →L[ℝ] C(K,F) := by
let L : C(K,E) →ₗ[ℝ] C(K,F) :=
{ toFun := fun u => ⟨fun t => A t (u t),A.continuous.clm_apply u.continuous⟩
map_add' := by intro u v; ext t; exact map_add (A t) (u t) (v t)
map_smul' := by intro c u; ext t; exact map_smul (A t) c (u t) }
exact L.mkContinuous ‖A‖ (by
intro u
change ‖(⟨fun t => A t (u t),A.continuous.clm_apply u.continuous⟩ : C(K,F))‖ ≤ ‖A‖ * ‖u‖
apply (ContinuousMap.norm_le _ (mul_nonneg (norm_nonneg A) (norm_nonneg u))).mpr
intro t
exact (A t).le_opNorm (u t) |>.trans
(mul_le_mul (A.norm_coe_le_norm t) (u.norm_coe_le_norm t)
(norm_nonneg _) (norm_nonneg _)))
@[simp] lemma pathLinearApply_apply (A : C(K,E →L[ℝ] F)) (u : C(K,E)) (t : K) :
pathLinearApply A u t = A t (u t) := rfl
lemma norm_pathLinearApply_le (A : C(K,E →L[ℝ] F)) : ‖pathLinearApply A‖ ≤ ‖A‖ := by
apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg A)
intro u
apply (ContinuousMap.norm_le _ (mul_nonneg (norm_nonneg A) (norm_nonneg u))).mpr
intro t
exact (A t).le_opNorm (u t) |>.trans
(mul_le_mul (A.norm_coe_le_norm t) (u.norm_coe_le_norm t)
(norm_nonneg _) (norm_nonneg _))
def pathLinearApplyL : C(K,E →L[ℝ] F) →L[ℝ] C(K,E) →L[ℝ] C(K,F) := by
let L : C(K,E →L[ℝ] F) →ₗ[ℝ] C(K,E) →L[ℝ] C(K,F) :=
{ toFun := fun (A : C(K,E →L[ℝ] F)) => pathLinearApply A
map_add' := by intro A B; ext u t; rfl
map_smul' := by intro c A; ext u t; rfl }
refine LinearMap.mkContinuous (𝕜 := ℝ) (𝕜₂ := ℝ)
(E := C(K,E →L[ℝ] F)) (F := C(K,E) →L[ℝ] C(K,F)) L 1 ?_
intro A
change ‖pathLinearApply A‖ ≤ 1 * ‖A‖
simpa only [one_mul] using norm_pathLinearApply_le A
lemma hasFDerivAt_path_comp (f : C(E,F)) (f' : C(E,E →L[ℝ] F))
(hder : ∀ x, HasFDerivAt f (f' x) x) (u : C(K,E)) :
HasFDerivAt (fun v : C(K,E) => f.comp v) (pathLinearApply (f'.comp u)) u := by
rw [hasFDerivAt_iff_isLittleO_nhds_zero,Asymptotics.isLittleO_iff]
intro ε hε
have hc : IsCompact (range u) := isCompact_range u.continuous
have huni := hc.uniformContinuousAt_of_continuousAt f'
(fun _ _ => f'.continuous.continuousAt)
(r := {z : (E →L[ℝ] F) × (E →L[ℝ] F) | dist z.1 z.2 < ε})
(Metric.mem_uniformity_dist.mpr ⟨ε,hε,fun {_ _} h => h⟩)
obtain ⟨δ,hδ,hclose⟩ := Metric.mem_uniformity_dist.mp huni
filter_upwards [Metric.ball_mem_nhds (0:C(K,E)) hδ] with v hv
have hvnorm : ‖v‖ < δ := by simpa using hv
apply (ContinuousMap.norm_le _ (mul_nonneg hε.le (norm_nonneg v))).mpr
intro t
have hbound : ∀ z ∈ Metric.ball (u t) δ, ‖f' z - f' (u t)‖ ≤ ε := by
intro z hz
have hh := hclose (by simpa only [Metric.mem_ball,dist_comm] using hz) (mem_range_self t)
change dist (f' (u t)) (f' z) < ε at hh
exact le_of_lt (by simpa only [dist_eq_norm, norm_sub_rev] using hh)
have hb : u t + v t ∈ Metric.ball (u t) δ := by
rw [Metric.mem_ball,dist_eq_norm,add_sub_cancel_left]
exact (v.norm_coe_le_norm t).trans_lt hvnorm
have hm := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le'
(fun z (_ : z ∈ Metric.ball (u t) δ) => (hder z).hasFDerivWithinAt)
hbound (convex_ball (u t) δ) (Metric.mem_ball_self hδ) hb
have heq : ((f.comp (u+v) - f.comp u - pathLinearApply (f'.comp u) v) t) =
f (u t + v t) - f (u t) - f' (u t) ((u t + v t) - u t) := by
simp only [ContinuousMap.sub_apply,ContinuousMap.comp_apply,ContinuousMap.add_apply,
pathLinearApply_apply,add_sub_cancel_left]
rw [heq]
exact hm.trans (by simpa only [add_sub_cancel_left] using
mul_le_mul_of_nonneg_left (v.norm_coe_le_norm t) hε.le)
lemma contDiff_path_comp_nat (m : ℕ) (f : C(E,F)) (hf : ContDiff ℝ m f) :
ContDiff ℝ m (fun v : C(K,E) => f.comp v) := by
induction m generalizing F with
| zero => exact contDiff_zero.mpr f.continuous_postcomp
| succ m ih =>
obtain ⟨f',hf',hder⟩ := contDiff_succ_iff_hasFDerivAt.mp hf
let F' : C(E,E →L[ℝ] F) := ⟨f',hf'.continuous⟩
apply contDiff_succ_iff_hasFDerivAt.mpr
refine ⟨fun v => pathLinearApply (F'.comp v),?_,?_⟩
· exact (pathLinearApplyL (K := K) (E := E) (F := F)).contDiff.comp
(ih F' hf')
· intro u; exact hasFDerivAt_path_comp f F' hder u
lemma contDiff_path_comp (f : C(E,F)) (hf : ContDiff ℝ ∞ f) :
ContDiff ℝ ∞ (fun v : C(K,E) => f.comp v) :=
contDiff_infty.mpr fun m => contDiff_path_comp_nat m f (contDiff_infty.mp hf m)
end WeakMTWTransport
end
end OAI
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