Theorems · Theorem · functional analysis
norm_pos_iff
∀ {E : Type u_5} [inst : NormedAddGroup E] {a : E}, 0 < ‖a‖ ↔ a ≠ 0- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 168 results in Mathlib
- Foundations
- Depth 115 from the axioms, rests on 2,433 definitions · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedAddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Norm.normstatement · cited by 5,413
- not_leproof · cited by 328
- NormedAddGroupstatement and proof · cited by 32
- norm_le_zero_iffproof · cited by 23
Cited by168
Results whose statement or proof uses this declaration.
- Complex.exp_logproof · cited by 14
- Complex.arg_real_mulproof · cited by 11
- NormedSpace.isVonNBounded_iffproof · cited by 9
- Complex.cos_argproof · cited by 7
- eq_zero_or_norm_posproof · cited by 6
- Complex.arg_mem_Iocproof · cited by 6
- Complex.arg_mul_coe_angleproof · cited by 6
- Seminorm.smul_ball_zeroproof · cited by 5
- tendsto_zero_of_isBoundedUnder_smul_of_tendsto_coboundedproof · cited by 5
- Orientation.oangle_addproof · cited by 5
- Complex.arg_nonneg_iffproof · cited by 4
- Function.locallyFinsuppWithin.logCounting_single_eq_log_sub_constproof · cited by 4