| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace KKTraceTransfer |
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| open Filter Topology |
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| noncomputable section |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
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| theorem bounded_evals_continuous {X : Type*} [TopologicalSpace X] |
| (F : X → (H →L[ℂ] H)) (hF : ∀ a, ‖F a‖ ≤ 1) |
| (D : ℕ → H) (hD : DenseRange D) (hc : ∀ j, Continuous (fun a => F a (D j))) |
| (ξ : H) : Continuous (fun a => F a ξ) := by |
| have hLip (a : X) (x y : H) : dist (F a x) (F a y) ≤ dist x y := by |
| rw [dist_eq_norm,dist_eq_norm,←map_sub] |
| exact ((F a).le_opNorm _).trans (by simpa using mul_le_mul_of_nonneg_right (hF a) (norm_nonneg (x-y))) |
| apply continuous_iff_continuousAt.mpr |
| intro a |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| obtain ⟨j, hj⟩ := hD.exists_dist_lt ξ (show 0 < ε/4 by positivity) |
| have hjc := Metric.tendsto_nhds.mp ( (hc j).continuousAt (x := a)) (ε/2) (show 0 < ε/2 by positivity) |
| filter_upwards [hjc] with b hb |
| calc |
| dist (F b ξ) (F a ξ) ≤ dist (F b ξ) (F b (D j)) + |
| (dist (F b (D j)) (F a (D j)) + dist (F a (D j)) (F a ξ)) := |
| (dist_triangle (F b ξ) (F b (D j)) (F a ξ)).trans |
| (add_le_add (le_refl _) (dist_triangle (F b (D j)) (F a (D j)) (F a ξ))) |
| _ ≤ dist ξ (D j) + (dist (F b (D j)) (F a (D j)) + dist (D j) ξ) := |
| add_le_add (hLip b _ _) (add_le_add (le_refl _) (hLip a _ _)) |
| _ < ε := by rw [dist_comm (D j) ξ]; linarith |
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| end |
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| end KKTraceTransfer |
| end |
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| end OAI |
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