Theorems · Theorem · functional analysis
gauge_add_le
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E},
Convex ℝ s → Absorbent ℝ s → ∀ (x y : E), gauge s (x + y) ≤ gauge s x + gauge s yGauge is subadditive.
- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 160 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.
Cites21
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
- Nat.cast_oneproof · cited by 2,501
- LT.lt.leproof · cited by 2,189
- Nat.cast_zeroproof · cited by 1,870
- le_of_ltproof · cited by 1,175
- Convexstatement and proof · cited by 551
- lt_of_not_geproof · cited by 374
- add_pos'proof · cited by 88
- gaugestatement and proof · cited by 85
Cited by5
Results whose statement or proof uses this declaration.
- gaugeSeminormproof · cited by 6
- continuousAt_gaugeproof · cited by 2
- separate_convex_open_setproof · cited by 2
- gauge_sum_leproof · cited by 1
- Convex.lipschitzWith_gaugeproof · cited by 1