| import OAI.Analysis.FiniteFactor.BoundedMatrixOperator |
| import Mathlib.Topology.MetricSpace.UniformConvergence |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open Filter |
| open scoped Topology |
| noncomputable section |
|
|
| /-- A uniformly bounded field is strongly continuous wherever all its basis |
| columns are continuous. No norm-continuity of the field is required. -/ |
| theorem fiberField_continuousOn {X : Type*} [TopologicalSpace X] |
| (A : X → SourceFiberHilbert →L[ℂ] SourceFiberHilbert) (Q : Set X) |
| (C : ℝ) (hC : ∀ x ∈ Q, ‖A x‖ ≤ C) |
| (hb : ∀ k : ℤ, ContinuousOn (fun x => A x (lp.single 2 k (1 : ℂ))) Q) |
| (v : SourceFiberHilbert) : ContinuousOn (fun x => A x v) Q := by |
| classical |
| let w : Finset ℤ → SourceFiberHilbert := fun s => ∑ k ∈ s, lp.single 2 k (v k) |
| have hw : Tendsto w atTop (𝓝 v) := lp.hasSum_single (by norm_num) v |
| have he : Tendsto (fun s => C * ‖v - w s‖) atTop (𝓝 0) := by |
| simpa using ((tendsto_const_nhds (x := v)).sub hw).norm.const_mul C |
| have hu : TendstoUniformlyOn (fun s x => A x (w s)) (fun x => A x v) atTop Q := by |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro ε hε |
| filter_upwards [(tendsto_order.mp he).2 ε hε] with s hs x hx |
| rw [dist_eq_norm, ← map_sub] |
| exact ((A x).le_opNorm (v - w s)).trans_lt |
| ((mul_le_mul_of_nonneg_right (hC x hx) (norm_nonneg _)).trans_lt hs) |
| apply hu.continuousOn |
| apply Filter.Eventually.frequently |
| apply Eventually.of_forall |
| intro s |
| have hh : (fun x => A x (w s)) = |
| fun x => ∑ k ∈ s, (v k) • A x (lp.single 2 k (1 : ℂ)) := by |
| funext x |
| simp only [w, map_sum] |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [show lp.single 2 k (v k) = (v k) • (lp.single 2 k (1 : ℂ) : SourceFiberHilbert) by |
| simp [← lp.single_smul], map_smul] |
| rw [hh] |
| exact continuousOn_finsetSum s (fun k _ => (hb k).const_smul (v k)) |
|
|
| /-- Evaluation on a varying continuous vector preserves strong continuity |
| under a uniform operator bound. -/ |
| theorem fiberField_continuousOn_apply {X : Type*} [TopologicalSpace X] |
| (A : X → SourceFiberHilbert →L[ℂ] SourceFiberHilbert) (Q : Set X) |
| (C : ℝ) (hC : ∀ x ∈ Q, ‖A x‖ ≤ C) |
| (hA : ∀ v, ContinuousOn (fun x => A x v) Q) |
| (v : X → SourceFiberHilbert) (hv : ContinuousOn v Q) : |
| ContinuousOn (fun x => A x (v x)) Q := by |
| intro x hx |
| have he : Tendsto (fun y => C * ‖v y - v x‖) (𝓝[Q] x) (𝓝 0) := by |
| simpa using ((hv x hx).sub_const (v x)).norm.const_mul C |
| have hzero : Tendsto (fun y => A y (v y - v x)) (𝓝[Q] x) (𝓝 0) := by |
| apply squeeze_zero_norm' _ he |
| filter_upwards [self_mem_nhdsWithin] with y hy |
| exact ((A y).le_opNorm _).trans |
| (mul_le_mul_of_nonneg_right (hC y hy) (norm_nonneg _)) |
| have ht := hzero.add (hA (v x) x hx) |
| change Tendsto _ _ _ |
| simpa only [map_sub, sub_add_cancel, zero_add] using ht |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
