Theorems · Theorem · functional analysis
mem_sphere_zero_iff_norm
∀ {E : Type u_5} [inst : SeminormedAddGroup E] {a : E} {r : ℝ}, a ∈ Metric.sphere 0 r ↔ ‖a‖ = r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext
- 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.spherestatement · cited by 371
- SeminormedAddGroupstatement and proof · cited by 331
- dist_zero_rightproof · cited by 172
- Metric.mem_sphereproof · cited by 9
Cited by13
Results whose statement or proof uses this declaration.
- norm_eq_of_mem_sphereproof · cited by 11
- Circle.norm_coeproof · cited by 5
- Unitary.spectrum_subset_circleproof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhdsproof · cited by 1
- AffineSpace.cobounded_eq_iSup_sphere_asymptoticNhdsproof · cited by 1
- Behrend.threeAPFree_sphereproof · cited by 1
- LinearMap.bound_of_sphere_boundproof · cited by 1
- ContinuousLinearMap.sSup_sphere_eq_nnnormproof · cited by 1
- StrictConvexSpace.of_norm_combo_ne_oneproof · cited by 1
- StrictConvexSpace.of_pairwise_sphere_norm_ne_twoproof · cited by 1
- Behrend.sphere_subset_preimage_metric_sphereproof · cited by 0
- StrictConvexSpace.of_norm_addproof · cited by 0