Theorems · Theorem · functional analysis
Absorbent.gauge_set_nonempty
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} {x : E},
Absorbent ℝ s → {r | 0 < r ∧ x ∈ r • s}.NonemptyIf the given subset is Absorbent then the set we take an infimum over in gauge is nonempty,
which is useful for proving many properties about the gauge.
- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Set.ofPredstatement and proof · cited by 6,101
- Norm.normproof · cited by 5,413
- Set.Nonemptystatement and proof · cited by 2,627
- LT.lt.leproof · cited by 2,189
- Set.smulSetstatement · cited by 608
- Eq.geproof · cited by 375
- Real.norm_of_nonnegproof · cited by 135
- Absorbentstatement and proof · cited by 60
Cited by2
Results whose statement or proof uses this declaration.
- exists_lt_of_gauge_ltproof · cited by 6
- gauge_monoproof · cited by 4