Theorems · Theorem · real analysis

hasFDerivWithinAt_piLp

∀ {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {H : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup H] [inst_2 : (i : ι) → NormedAddCommGroup (E i)] [inst_3 : (i : ι) → NormedSpace 𝕜 (E i)]
  [inst_4 : NormedSpace 𝕜 H] [Finite ι] (p : ENNReal) [Fact (1 ≤ p)] {f : H → PiLp p E} {f' : H →L[𝕜] PiLp p E}
  {t : Set H} {y : H},
  HasFDerivWithinAt f f' t y ↔ ∀ (i : ι), HasFDerivWithinAt (fun x => (f x).ofLp i) (PiLp.proj p E i ∘SL f') t y

Defined in

Mathlib.Analysis.Calculus.FDeriv.WithLp

Cited by

1 results in Mathlib

Foundations

Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceFiniteFact