Theorems · Theorem · real analysis

eVariationOn_bilinear_comp_le

∀ {α : Type u_2} [inst : LinearOrder α] {E : Type u_3} {F : Type u_4} {G : Type u_5} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] [inst_5 : NormedAddCommGroup G]
  [inst_6 : NormedSpace ℝ G] {s : Set α} {f : α → E} {g : α → F} {C D : ENNReal},
  (∀ x ∈ s, ‖f x‖ₑ ≤ C) →
    (∀ x ∈ s, ‖g x‖ₑ ≤ D) →
      ∀ (B : E →L[ℝ] F →L[ℝ] G),
        eVariationOn (fun x => (B (f x)) (g x)) s ≤ ‖B‖ₑ * (C * eVariationOn g s + D * eVariationOn f s)

Defined in

Mathlib.Analysis.BoundedVariation

Cited by

2 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace