File size: 2,917 Bytes
ddc84c5 | 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 | 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
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