Theorems · Theorem · functional analysis
ContinuousLinearMap.opNorm_lsmul
∀ (𝕜 : Type u_1) {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] (R : Type u_3) [inst_3 : NormedDivisionRing R] [inst_4 : NormedAlgebra 𝕜 R]
[inst_5 : Module R E] [inst_6 : NormSMulClass R E] [inst_7 : IsScalarTower 𝕜 R E] [Nontrivial E],
‖ContinuousLinearMap.lsmul 𝕜 R‖ = 1The norm of lsmul equals 1 in any nontrivial normed group.
This is ContinuousLinearMap.opNorm_lsmul_le as an equality.
- Defined in
- Mathlib.Analysis.Normed.Operator.Mul
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Realstatement and proof · cited by 25,697
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement and proof · cited by 5,413
- ContinuousLinearMapstatement · cited by 5,352
- IsScalarTowerstatement and proof · cited by 3,896
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
Cited by1
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.opNNNorm_lsmulproof · cited by 1