Theorems · Theorem · functional analysis
mem_closedBall_zero_iff
∀ {E : Type u_5} [inst : SeminormedAddGroup E] {a : E} {r : ℝ}, a ∈ Metric.closedBall 0 r ↔ ‖a‖ ≤ r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 97 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.closedBallstatement · cited by 704
- SeminormedAddGroupstatement and proof · cited by 331
- dist_zero_rightproof · cited by 172
- Metric.mem_closedBallproof · cited by 42
Cited by16
Results whose statement or proof uses this declaration.
- isBounded_iff_forall_norm_leproof · cited by 17
- ZLattice.rankproof · cited by 5
- Complex.UnitClosedDisc.norm_le_oneproof · cited by 4
- mem_closure_zero_iff_normproof · cited by 3
- div_le_egauge_closedBallproof · cited by 2
- unique_topology_of_t2proof · cited by 1
- eventually_singleton_add_smul_subsetproof · cited by 1
- spectrum.differentiableOn_inverse_one_sub_smulproof · cited by 1
- NormedSpace.closedBall_inv_subset_polar_closedBallproof · cited by 1
- MeasureTheory.hasFDerivAt_convolution_right_with_paramproof · cited by 1
- ContinuousLinearMap.sSup_unitClosedBall_eq_nnnormproof · cited by 1
- Complex.CanonicalDecomp.divisor_eq_divisorproof · cited by 1