Theorems · Theorem · functional analysis
NormedSpace.isBounded_iff_subset_smul_closedBall
∀ (𝕜 : Type u_1) {E : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {s : Set E}, Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1- Defined in
- Mathlib.Analysis.LocallyConvex.Bounded
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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 · cited by 25,697
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- LE.le.transproof · cited by 3,151
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Metric.ballproof · cited by 735
- Metric.closedBallstatement and proof · cited by 704
- Set.smulSetstatement · cited by 608
- Bornology.IsBoundedstatement · cited by 293
- Metric.ball_subset_closedBallproof · cited by 46
- Set.smul_set_monoproof · cited by 15
Cited by1
Results whose statement or proof uses this declaration.
- BoxIntegral.integrable_of_continuousOnproof · cited by 3