Theorems · Theorem · real analysis

contDiff_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] [inst_5 : Fintype ι] (p : ENNReal) [inst_6 : Fact (1 ≤ p)] {n : WithTop ℕ∞}
  {f : H → PiLp p E}, ContDiff 𝕜 n f ↔ ∀ (i : ι), ContDiff 𝕜 n fun x => (f x).ofLp i

Defined in

Mathlib.Analysis.Calculus.ContDiff.WithLp

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceFintypeFact