Theorems · Theorem · functional analysis
gauge_le_one_iff_mem_closure
∀ {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 → (gauge s x ≤ 1 ↔ x ∈ closure s)- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Filterstatement · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- IsTopologicalAddGroupstatement and proof · cited by 1,394
- closurestatement and proof · cited by 1,254
- ContinuousSMulstatement and proof · cited by 1,016
- Convexstatement and proof · cited by 551
- Continuous.continuousOnproof · cited by 311
Cited by2
Results whose statement or proof uses this declaration.
- mapsTo_gaugeRescale_closureproof · cited by 1
- gauge_eq_one_iff_mem_frontierproof · cited by 0