Theorems · Theorem · real analysis

ContDiffWithinAt.smul_const

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : WithTop ℕ∞}
  {A : Type u_4} [inst_5 : NormedRing A] [inst_6 : NormedAlgebra 𝕜 A] [inst_7 : Module A F] [IsScalarTower 𝕜 A F]
  [IsBoundedSMul A F] {s : Set E} {f : E → A} {x : E},
  ContDiffWithinAt 𝕜 n f s x → ∀ (v : F), ContDiffWithinAt 𝕜 n (fun y => f y • v) s x

The scalar multiplication of C^n function within a set at a point and a constant and is C^n within this set at this point.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedRingNormedAlgebraModuleIsScalarTowerIsBoundedSMul