Theorems · Theorem · functional analysis
balanced_iff_smul_mem
∀ {𝕜 : Type u_1} {E : Type u_3} [inst : SeminormedRing 𝕜] [inst_1 : SMul 𝕜 E] {s : Set E},
Balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s- Defined in
- Mathlib.Analysis.LocallyConvex.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedRingSMul
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Norm.normstatement and proof · cited by 5,413
- SeminormedRingstatement and proof · cited by 446
- Balancedstatement · cited by 77
- Set.smul_set_subset_iffproof · cited by 7
Cited by2
Results whose statement or proof uses this declaration.
- Balanced.smul_memproof · cited by 7
- Balanced.convexHullproof · cited by 2