contDiffOn_tsum_cexp — Mathlib

English

The norm of a finite-support vector equals the maximum of the absolute values of its coordinates.

Русский

Норма конечномерного вектора равна максимуму модулей его компонент.

LaTeX

$$‖x‖ = max(x0.natAbs, x1.natAbs)$$

Lean4

theorem contDiffOn_tsum_cexp (k : ℕ∞) : ContDiffOn ℂ k (fun z : ℂ ↦ ∑' n : ℕ, cexp (2 * π * I * z) ^ n) ℍₒ :=
  contDiffOn_of_differentiableOn_deriv fun m _ z hz ↦
    (((summableLocallyUniformlyOn_iteratedDerivWithin_cexp m).differentiableOn isOpen_upperHalfPlaneSet
            (fun n _ hz ↦ differentiableAt_iteratedDerivWithin_cexp n m isOpen_upperHalfPlaneSet hz))
          z hz).congr
      (fun z hz ↦ iteratedDerivWithin_tsum_cexp_eq m ⟨z, hz⟩) (iteratedDerivWithin_tsum_cexp_eq m ⟨z, hz⟩)

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