Theorems · Theorem · real analysis

eVariationOn_smul_le

∀ {α : Type u_2} [inst : LinearOrder α] {F : Type u_4} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace ℝ F]
  {𝕜 : Type u_6} {f : α → 𝕜} {g : α → F} [inst_3 : NormedRing 𝕜] [inst_4 : NormedAlgebra ℝ 𝕜] [inst_5 : Module 𝕜 F]
  [NormSMulClass 𝕜 F] [IsScalarTower ℝ 𝕜 F] {C D : ENNReal} {s : Set α},
  (∀ x ∈ s, ‖f x‖ₑ ≤ C) → (∀ x ∈ s, ‖g x‖ₑ ≤ D) → eVariationOn (f • g) s ≤ C * eVariationOn g s + D * eVariationOn f s

Defined in

Mathlib.Analysis.BoundedVariation

Cited by

2 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderNormedAddCommGroupNormedSpaceNormedRingNormedAlgebraModuleNormSMulClassIsScalarTower