| import Mathlib.Analysis.LocallyConvex.WeakSpace | |
| import Mathlib.Analysis.Normed.Module.DoubleDual | |
| import Mathlib.Analysis.InnerProductSpace.Dual | |
| namespace OAI | |
| namespace CompanionFPP | |
| open Topology Filter Metric Bornology | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] | |
| lemma hilbert_bidual_surjective : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ H) := by | |
| intro F | |
| let g : StrongDual ℝ H := F.comp (InnerProductSpace.toDual ℝ H).toContinuousLinearEquiv.toContinuousLinearMap | |
| refine ⟨(InnerProductSpace.toDual ℝ H).symm g, ?_⟩ | |
| ext l | |
| obtain ⟨y, rfl⟩ := (InnerProductSpace.toDual ℝ H).surjective l | |
| change inner ℝ y ((InnerProductSpace.toDual ℝ H).symm g) = g y | |
| rw [real_inner_comm, InnerProductSpace.toDual_symm_apply] | |
| omit [CompleteSpace H] in | |
| lemma isClosed_weakImage {s : Set H} (hc : Convex ℝ s) (hs : IsClosed s) : | |
| IsClosed (toWeakSpace ℝ H '' s) := by | |
| rw [← closure_eq_iff_isClosed, ← hc.toWeakSpace_closure ℝ, hs.closure_eq] | |
| theorem hilbert_isCompact_weak_closedBall (r : ℝ) : | |
| IsCompact (toWeakSpace ℝ H '' closedBall (0 : H) r) := by | |
| have hc := NormedSpace.isCompact_closure_of_isBounded ℝ H | |
| (toWeakSpace ℝ H '' closedBall (0 : H) r) (by | |
| simpa only [Set.preimage_image_eq _ (toWeakSpace ℝ H).injective] using | |
| (isBounded_closedBall (x := (0 : H)) (r := r))) (by | |
| intro x _ | |
| obtain ⟨y, hy⟩ := hilbert_bidual_surjective (H := H) (WeakDual.toStrongDual x) | |
| refine ⟨toWeakSpace ℝ H y, ?_⟩ | |
| exact hy) | |
| rwa [(isClosed_weakImage (convex_closedBall (0 : H) r) isClosed_closedBall).closure_eq] at hc | |
| theorem exists_weak_product_limit {I : Type*} (f : ℕ → I → H) (R : I → ℝ) | |
| (hb : ∀ n i, ‖f n i‖ ≤ R i) : | |
| ∃ (u : I → WeakSpace ℝ H) (l : Filter ℕ), l.NeBot ∧ l ≤ atTop ∧ | |
| Tendsto (fun n i => toWeakSpace ℝ H (f n i)) l (𝓝 u) := by | |
| have hc : IsCompact {x : I → WeakSpace ℝ H | ∀ i, x i ∈ toWeakSpace ℝ H '' closedBall (0 : H) (R i)} := | |
| isCompact_pi_infinite (fun i => hilbert_isCompact_weak_closedBall (H := H) (R i)) | |
| obtain ⟨u, _, hu⟩ := hc.exists_mapClusterPt (f := atTop) | |
| (u := fun n i => toWeakSpace ℝ H (f n i)) (le_principal_iff.mpr (by | |
| change ∀ᶠ n : ℕ in atTop, ∀ i, toWeakSpace ℝ H (f n i) ∈ _ | |
| exact Eventually.of_forall (fun n i => ⟨f n i, mem_closedBall_zero_iff.mpr (hb n i), rfl⟩))) | |
| exact ⟨u, atTop ⊓ comap (fun n i => toWeakSpace ℝ H (f n i)) (𝓝 u), | |
| (by simpa only [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, inf_comm] using hu), inf_le_left, tendsto_iff_comap.mpr inf_le_right⟩ | |
| end CompanionFPP | |
| end OAI | |