Theorems · Theorem · functional analysis

egauge_le_of_smul_mem_of_ne

∀ {𝕜 : Type u_1} [inst : NormedDivisionRing 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E] {c : 𝕜}
  {s : Set E} {x : E}, c • x ∈ s → c ≠ 0 → egauge 𝕜 s x ≤ ↑‖c‖₊⁻¹

If c • x ∈ s and c ≠ 0, then egauge 𝕜 s x is at most (‖c‖₊⁻¹ : ℝ≥0). See also egauge_le_of_smul_mem.

Defined in

Mathlib.Analysis.Convex.EGauge

Cited by

2 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedDivisionRingAddCommGroupModule