Theorems · Theorem · general topology
AddMonoidHomClass.isometry_iff_norm
∀ {𝓕 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : SeminormedAddGroup E] [inst_1 : SeminormedAddGroup F]
[inst_2 : FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕), Isometry ⇑f ↔ ∀ (x : E), ‖f x‖ = ‖x‖- Defined in
- Mathlib.Analysis.Normed.Group.Uniform
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- add_zeroproof · cited by 2,707
- FunLikestatement and proof · cited by 2,560
- map_addproof · cited by 964
- map_negproof · cited by 378
- SeminormedAddGroupstatement and proof · cited by 331
- AddMonoidHomClassstatement and proof · cited by 252
- Isometrystatement · cited by 230
- norm_negproof · cited by 190
- dist_eq_norm_neg_addproof · cited by 46
Cited by5
Results whose statement or proof uses this declaration.
- AddMonoidHomClass.isometry_of_normproof · cited by 18
- ContinuousLinearMap.opNorm_mul_applyproof · cited by 3
- Unitization.antilipschitzWith_addEquivproof · cited by 2
- NormedAddGroupHom.norm_eq_of_isometryproof · cited by 1
- ContinuousLinearMap.isometry_iff_adjoint_comp_selfproof · cited by 1