Theorems · Theorem · functional analysis
mem_closedBall_iff_norm
∀ {E : Type u_5} [inst : SeminormedAddCommGroup E] {a b : E} {r : ℝ}, b ∈ Metric.closedBall a r ↔ ‖b - a‖ ≤ r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
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
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Metric.closedBallstatement · cited by 704
- Metric.mem_closedBallproof · cited by 42
- dist_eq_norm_subproof · cited by 29
Cited by12
Results whose statement or proof uses this declaration.
- MeasureTheory.addHaar_image_le_mul_of_det_ltproof · cited by 4
- ContDiffBump.one_of_mem_closedBallproof · cited by 4
- ODE.FunSpace.mem_closedBallproof · cited by 3
- ApproximatesLinearOn.norm_fderiv_sub_leproof · cited by 3
- ContinuousLinearMap.is_weak_closed_closedBallproof · cited by 3
- InnerProductSpace.HarmonicContOnCl.circleAverage_eqproof · cited by 1
- sub_isBigO_norm_rpow_add_one_of_fderivproof · cited by 1
- IsPicardLindelof.of_contDiffAt_oneproof · cited by 1
- setOf_sub_mem_closedBall_eq_closedBallproof · cited by 0
- add_mem_closedBall_iff_normproof · cited by 0
- lp.tendsto_lp_of_tendsto_piproof · cited by 0