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67aee1d | 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 | import Mathlib
namespace OAI
noncomputable section
namespace KKTraceTransfer
open Filter Topology
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
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
end
end KKTraceTransfer
end
end OAI
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