Theorems · Theorem · functional analysis
mem_ball_zero_iff
∀ {E : Type u_5} [inst : SeminormedAddGroup E] {a : E} {r : ℝ}, a ∈ Metric.ball 0 r ↔ ‖a‖ < r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- Metric.ballstatement · cited by 735
- SeminormedAddGroupstatement and proof · cited by 331
- dist_zero_rightproof · cited by 172
- Metric.mem_ballproof · cited by 47
Cited by22
Results whose statement or proof uses this declaration.
- Complex.UnitDisc.norm_lt_oneproof · cited by 7
- MeasureTheory.convolution_eq_right'proof · cited by 4
- egauge_ball_le_of_one_lt_normproof · cited by 4
- norm_add_lt_of_not_sameRayproof · cited by 3
- UpperHalfPlane.differentiableOn_cuspFunction_ballproof · cited by 3
- ModularForm.tendsto_atImInfty_tprod_one_sub_eta_q_powproof · cited by 3
- ContinuousLinearMap.sSup_unit_ball_eq_nnnormproof · cited by 2
- norm_sum_lt_of_strictConvexSpaceproof · cited by 1
- OpenPartialHomeomorph.contDiffOn_univUnitBall_symmproof · cited by 1
- MeasureTheory.dist_convolution_le'proof · cited by 1
- Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zeroproof · cited by 1
- tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_auxproof · cited by 1