Theorems · Theorem · global analysis

HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type v} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  {p : FormalMultilinearSeries 𝕜 E F} {r : ENNReal} {f : E → F} {x : E} {s : Set E},
  HasFPowerSeriesWithinOnBall f p s x r →
    ∀ {f' : E →L[𝕜] F} {y : E},
      ↑‖y‖₊ < r →
        x + y ∈ insert x s →
          HasFDerivWithinAt f f' (insert x s) (x + y) →
            UniqueDiffOn 𝕜 (insert x s) → HasSum (fun n => (p.derivSeries n) fun x => y) f'

If a function has a power series p within a set of unique differentiability, inside a ball, and is differentiable at a point, then the derivative series of p is summable at a point, with sum the given differential. Note that this theorem does not require completeness of the space.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

1 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace