Theorems · Theorem · real analysis

ContDiff.const_smul

∀ {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∞}
  {R : Type u_3} [inst_5 : DistribSMul R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {f : E → F} (c : R),
  ContDiff 𝕜 n f → ContDiff 𝕜 n fun y => c • f y

The scalar multiplication of a constant and a C^n function is C^n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

4 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceDistribSMulSMulCommClassContinuousConstSMul