Theorems · Theorem · general topology

MonoidHomClass.lipschitz_of_bound

∀ {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : SeminormedGroup E] [inst_1 : SeminormedGroup F]
  [inst_2 : FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ),
  (∀ (x : E), ‖f x‖ ≤ C * ‖x‖) → LipschitzWith C.toNNReal ⇑f

A homomorphism f of seminormed groups is Lipschitz, if there exists a constant C such that for all x, one has ‖f x‖ ≤ C * ‖x‖. The analogous condition for a linear map of (semi)normed spaces is in Mathlib/Analysis/Normed/Operator/Basic.lean.

Defined in

Mathlib.Analysis.Normed.Group.Uniform

Cited by

3 results in Mathlib

Foundations

Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedGroupSeminormedGroupFunLikeMonoidHomClass