Theorems · Theorem · functional analysis
Balanced.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 ∈ sAlias of the forward direction of balanced_iff_smul_mem.
- Defined in
- Mathlib.Analysis.LocallyConvex.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 97 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 · cited by 5,413
- SeminormedRingstatement and proof · cited by 446
- Balancedstatement · cited by 77
- balanced_iff_smul_memproof · cited by 2
Cited by7
Results whose statement or proof uses this declaration.
- Balanced.balancedHull_subset_of_subsetproof · cited by 3
- Balanced.neg_mem_iffproof · cited by 2
- ContinuousSMul.topology_eq_of_nhds_inf_principal_eqproof · cited by 1
- Balanced.zero_insert_interiorproof · cited by 1
- balancedCoreAux_maximalproof · cited by 1
- unique_topology_of_t2proof · cited by 1
- Balanced.smul_mem_monoproof · cited by 1