Theorems · Theorem · functional analysis
RingHomIsometric.norm_map
∀ {R₁ : Type u_5} {R₂ : Type u_6} {inst : Semiring R₁} {inst_1 : Semiring R₂} {inst_2 : Norm R₁} {inst_3 : Norm R₂}
{σ : R₁ →+* R₂} [self : RingHomIsometric σ] {x : R₁}, ‖σ x‖ = ‖x‖The ring homomorphism is an isometry.
- Defined in
- Mathlib.Analysis.Normed.Ring.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses no axioms
- Assumes
- RingHomIsometric
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.
- DFunLike.coestatement · cited by 62,936
- Realstatement · cited by 25,697
- Semiringstatement and proof · cited by 13,802
- RingHomstatement and proof · cited by 10,189
- Norm.normstatement · cited by 5,413
- Normstatement and proof · cited by 512
- RingHomIsometricstatement and proof · cited by 282
Cited by9
Results whose statement or proof uses this declaration.
- RingHom.isometryproof · cited by 4
- RingHomIsometric.nnnorm_mapproof · cited by 3
- ContinuousLinearMap.opNorm_le_of_unit_normproof · cited by 2
- RingHomIsometric.invproof · cited by 2
- ContinuousLinearMap.exists_approx_preimage_norm_leproof · cited by 1
- LinearMap.continuousAt_zero_of_locally_boundedproof · cited by 1
- LinearMap.antilipschitz_of_comap_nhds_leproof · cited by 1
- ContinuousLinearMapWOT.continuous_postcompproof · cited by 0
- IsCompactOperator.continuousproof · cited by 0