Theorems · Theorem · functional analysis
mem_closure_of_gauge_le_one
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} {x : E} [inst_2 : TopologicalSpace E]
[ContinuousSMul ℝ E], Convex ℝ s → 0 ∈ s → Absorbent ℝ s → gauge s x ≤ 1 → x ∈ closure s- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
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
- Filter.Eventuallyproof · cited by 3,134
- nhdsWithinproof · cited by 1,912
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- one_smulproof · cited by 1,374
- closurestatement · cited by 1,254
- Set.Iioproof · cited by 1,166
Cited by3
Results whose statement or proof uses this declaration.
- gauge_le_one_iff_mem_closureproof · cited by 2
- mapsTo_gaugeRescale_closureproof · cited by 1
- mem_frontier_of_gauge_eq_oneproof · cited by 0