Theorems · Theorem · real analysis

HasDerivAt.comp_semilinear

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {σ : 𝕜 →+* 𝕜} (σ' : 𝕜 →+* 𝕜) [RingHomIsometric σ]
  [RingHomInvPair σ σ'] {F' : Type u_1} [inst_5 : NormedAddCommGroup F'] [inst_6 : NormedSpace 𝕜 F'] (L : F →SL[σ] F'),
  HasDerivAt f f' x → HasDerivAt (⇑L ∘ f ∘ ⇑σ') (L f') (σ x)

If L is a σ-semilinear map, and f has Fréchet derivative f' at x, then L ∘ f ∘ σ⁻¹ has Fréchet derivative L ∘ f' at σ x.

Defined in

Mathlib.Analysis.Calculus.Deriv.Basic

Cited by

2 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceRingHomIsometricRingHomInvPairNormedAddCommGroupNormedSpace