Theorems · Theorem · real analysis

hasDerivAt_tsum_of_isPreconnected

∀ {α : Type u_1} {𝕜 : Type u_3} {F : Type u_5} [inst : NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]
  [inst_2 : NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [inst_4 : NormedSpace 𝕜 F] {g g' : α → 𝕜 → F}
  {t : Set 𝕜} {y₀ y : 𝕜},
  Summable u →
    IsOpen t →
      IsPreconnected t →
        (∀ (n : α), ∀ y ∈ t, HasDerivAt (g n) (g' n y) y) →
          (∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n) →
            y₀ ∈ t → (Summable fun n => g n y₀) → y ∈ t → HasDerivAt (fun z => ∑' (n : α), g n z) (∑' (n : α), g' n y) y

Consider a series of functions ∑' n, f n x on a preconnected open set. If the series converges at a point, and all functions in the series are differentiable with a summable bound on the derivatives, then the series is differentiable on the set and its derivative is the sum of the derivatives.

Defined in

Mathlib.Analysis.Calculus.SmoothSeries

Cited by

1 results in Mathlib

Foundations

Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldIsRCLikeNormedFieldNormedAddCommGroupCompleteSpaceNormedSpace