Theorems · Theorem · general topology

AddMonoidHomClass.continuous_of_bound

∀ {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : SeminormedAddGroup E] [inst_1 : SeminormedAddGroup F]
  [inst_2 : FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ), (∀ (x : E), ‖f x‖ ≤ C * ‖x‖) → Continuous ⇑f

A homomorphism f of seminormed groups is continuous, if there exists a constant C such that for all x, one has ‖f x‖ ≤ C * ‖x‖.

Defined in

Mathlib.Analysis.Normed.Group.Uniform

Cited by

2 results in Mathlib

Foundations

Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddGroupSeminormedAddGroupFunLikeAddMonoidHomClass