Theorems · Theorem · real analysis

norm_curveIntegral_segment_le

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {a b : E} {ω : E → E →L[𝕜] F}
  [inst_5 : NormedSpace ℝ E] {C : ℝ},
  (∀ z ∈ segment ℝ a b, ‖ω z‖ ≤ C) → ‖∫ᶜ (x : E) in Path.segment a b, ω x‖ ≤ C * ‖b - a‖

If ‖ω z‖ ≤ C at all points of the segment [a -[ℝ] b], then the curve integral ∫ᶜ x in .segment a b, ω x has norm at most C * ‖b - a‖.

Defined in

Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Cited by

1 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpace