Theorems · Theorem · functional analysis

continuousAt_gauge

∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} {x : E} [inst_2 : TopologicalSpace E]
  [IsTopologicalAddGroup E] [ContinuousSMul ℝ E], Convex ℝ s → s ∈ nhds 0 → ContinuousAt (gauge s) x

If s is a convex neighborhood of the origin in a topological real vector space, then gauge s is continuous. If the ambient space is a normed space, then gauge s is Lipschitz continuous, see Convex.lipschitz_gauge.

Defined in

Mathlib.Analysis.Convex.Gauge

Cited by

2 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousSMul