Theorems · Theorem · functional analysis
NormedAddGroupHom.le_opNorm
∀ {V₁ : Type u_2} {V₂ : Type u_3} [inst : SeminormedAddCommGroup V₁] [inst_1 : SeminormedAddCommGroup V₂]
(f : NormedAddGroupHom V₁ V₂) (x : V₁), ‖f x‖ ≤ ‖f‖ * ‖x‖The fundamental property of the operator norm: ‖f x‖ ≤ ‖f‖ * ‖x‖.
- Defined in
- Mathlib.Analysis.Normed.Group.Hom
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 117 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 and proof · cited by 25,697
- Set.ofPredproof · cited by 6,101
- Norm.normstatement and proof · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- MulZeroClass.mul_zeroproof · cited by 2,091
- norm_nonnegproof · cited by 725
- lt_of_le_of_neproof · cited by 230
- NormedAddGroupHomstatement and proof · cited by 216
- div_le_iff₀proof · cited by 97
- le_csInfproof · cited by 36
- NormedAddGroupHom.boundproof · cited by 2
Cited by9
Results whose statement or proof uses this declaration.
- NormedAddGroupHom.norm_completionproof · cited by 2
- NormedAddGroupHom.le_of_opNorm_leproof · cited by 2
- SeparationQuotient.norm_liftNormedAddGroupHom_apply_leproof · cited by 2
- QuotientAddGroup.norm_lift_apply_leproof · cited by 2
- NormedAddGroupHom.lipschitzproof · cited by 1
- NormedAddGroupHom.ratio_le_opNormproof · cited by 1
- NormedAddGroupHom.ker_completionproof · cited by 0
- NormedAddGroupHom.le_opNorm_of_leproof · cited by 0
- NormedAddGroupHom.opNorm_zero_iffproof · cited by 0