| import Mathlib |
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| namespace OAI |
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| 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] |
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| 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 |
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| end |
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| 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) |
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| def compactSmoothJetMap : 𝓓_{K}(E,F) →L[ℝ] ((m : ℕ) → E →ᵇ (E [×m]→L[ℝ] F)) := |
| ContinuousLinearMap.pi (ContDiffMapSupportedIn.structureMapCLM ℝ ⊤) |
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| 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) |
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| variable [CompleteSpace F] |
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| 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 |
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| end YauCounterexamples |
| end |
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| end OAI |
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