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695bd70 | 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 | import Mathlib
namespace OAI
noncomputable section
namespace YauCounterexamples
section
open Set Filter Function Metric TopologicalSpace
open scoped Topology ContDiff Distributions BoundedContinuousFunction
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem smooth_of_uniform_jets (u : ℕ → E → F) (hu : ∀ j, ContDiff ℝ ∞ (u j))
(G : (m : ℕ) → E →ᵇ (E [×m]→L[ℝ] F))
(hG : ∀ m, TendstoUniformly (fun j => iteratedFDeriv ℝ m (u j)) (G m) atTop) :
∃ f : E → F, ContDiff ℝ ∞ f ∧
∀ m, iteratedFDeriv ℝ m f = G m := by
let f : E → F := fun x => (G 0 x).curry0
let p : E → FormalMultilinearSeries ℝ E F := fun x m => G m x
have hp : HasFTaylorSeriesUpTo ∞ f p := by
refine ⟨fun _ => rfl, ?_, fun m _ => (G m).continuous⟩
intro m hm x
have ht : TendstoUniformly
(fun j y => (iteratedFDeriv ℝ (m+1) (u j) y).curryLeft)
(fun y => (G (m+1) y).curryLeft) atTop := by
exact (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (m+1) => E) F).isometry.uniformContinuous
|>.comp_tendstoUniformly (hG (m+1))
exact hasFDerivAt_of_tendstoUniformly ht
(fun j y => (hu j).ftaylorSeries.fderiv m hm y)
(fun y => (hG m).tendsto_at y) x
refine ⟨f,hp.contDiff,fun m => ?_⟩
funext x
exact (hp.eq_iteratedFDeriv (m:=m) (ENat.natCast_le_of_coe_top_le_withTop le_rfl m) x).symm
end
open Set Filter Function Metric TopologicalSpace
open scoped Topology ContDiff Distributions BoundedContinuousFunction
open Set Filter Function Metric TopologicalSpace
open scoped Topology ContDiff Distributions BoundedContinuousFunction
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] (K : Compacts E)
def compactSmoothJetMap : 𝓓_{K}(E,F) →L[ℝ] ((m : ℕ) → E →ᵇ (E [×m]→L[ℝ] F)) :=
ContinuousLinearMap.pi (ContDiffMapSupportedIn.structureMapCLM ℝ ⊤)
theorem compactSmoothJetMap_isClosed : IsClosed (range (compactSmoothJetMap (F:=F) K)) := by
let : ∀ m : ℕ, FirstCountableTopology (E →ᵇ (E [×m]→L[ℝ] F)) := fun _ => inferInstance
let : FirstCountableTopology ((m : ℕ) → E →ᵇ (E [×m]→L[ℝ] F)) := inferInstance
apply IsSeqClosed.isClosed
intro v G hv hlim
choose u hu using hv
have hG : ∀ m, TendstoUniformly (fun j => iteratedFDeriv ℝ m (u j)) (G m) atTop := by
intro m
have ht := (continuous_apply m).tendsto G |>.comp hlim
have htu : Tendsto (fun j => ContDiffMapSupportedIn.structureMapCLM ℝ ⊤ m (u j))
atTop (𝓝 (G m)) := by
convert ht using 1
funext j
exact congrArg (fun w => w m) (hu j)
simpa only [ContDiffMapSupportedIn.structureMapCLM_top_apply] using
BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp htu
obtain ⟨f,hf,hfj⟩ := smooth_of_uniform_jets (fun j => u j) (fun j => (u j).contDiff) G hG
have hzero : EqOn f 0 Kᶜ := by
intro x hx
have hconst : ∀ j, iteratedFDeriv ℝ 0 (u j) x = 0 := by
intro j
ext z
simp only [iteratedFDeriv_zero_apply]
exact (u j).zero_on_compl hx
have hGX : G 0 x = 0 := tendsto_nhds_unique ((hG 0).tendsto_at x)
(by simpa only [hconst] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : E [×0]→L[ℝ] F)) atTop (𝓝 0)))
have hh := congrArg (fun h : E → E [×0]→L[ℝ] F => h x) (hfj 0)
rw [hGX] at hh
have he := congrArg (fun p : E [×0]→L[ℝ] F => p (fun _ => 0)) hh
simpa using he
refine ⟨⟨f,hf,hzero⟩,?_⟩
funext m
apply BoundedContinuousFunction.ext
intro x
rw [compactSmoothJetMap, ContinuousLinearMap.pi_apply,
ContDiffMapSupportedIn.structureMapCLM_top_apply]
exact congrArg (fun h : E → E [×m]→L[ℝ] F => h x) (hfj m)
variable [CompleteSpace F]
theorem compactSmooth_complete : CompleteSpace 𝓓_{K}(E,F) := by
apply (ContDiffMapSupportedIn.isUniformEmbedding_pi_structureMapCLM (𝕜:=ℝ)
(n:=⊤) (K:=K)).isUniformInducing.completeSpace
exact (compactSmoothJetMap_isClosed (F:=F) K).isComplete
end YauCounterexamples
end
end OAI
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