Theorems · Theorem · functional analysis
ContinuousLinearMap.opNorm_mul_apply_le
∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] (R : Type u_3) [inst_1 : NonUnitalSeminormedRing R] [inst_2 : NormedSpace 𝕜 R] [inst_3 : IsScalarTower 𝕜 R R] [inst_4 : SMulCommClass 𝕜 R R] (x : R), ‖(ContinuousLinearMap.mul 𝕜 R) x‖ ≤ ‖x‖
- Defined in
- Mathlib.Analysis.Normed.Operator.Mul
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- RingHom.idstatement · cited by 18,349
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement · cited by 5,413
- ContinuousLinearMapstatement · cited by 5,352
- IsScalarTowerstatement and proof · cited by 3,896
- SMulCommClassstatement and proof · cited by 1,927
- norm_nonnegproof · cited by 725
- ContinuousLinearMap.mulstatement and proof · cited by 63
- ContinuousLinearMap.opNorm_le_boundproof · cited by 59
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.opNorm_mulLeftRight_apply_apply_leproof · cited by 1
- Unitization.norm_splitMul_snd_sqproof · cited by 0