compContinuousLinearMap — Mathlib

English

If the n-th norm of p_n grows no faster than a fixed constant times r^n, then r is a lower bound for the radius: r ≤ radius(p).

Русский

Если ‖p_n‖ ≤ C r^{-n} для некоторой константы C, то радиус по p не меньше r.

LaTeX

$$(∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C) → AnalyticOnNhd 𝕜 (N.prod fun n => f n) s$$

Lean4

theorem compContinuousLinearMap (hf : AnalyticWithinAt 𝕜 f s (u x)) : AnalyticWithinAt 𝕜 (f ∘ u) (u ⁻¹' s) x :=
  let ⟨p, hp⟩ := hf
  ⟨p.compContinuousLinearMap u, hp.compContinuousLinearMap⟩

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