Theorems · Theorem · functional analysis
ContinuousLinearMap.le_of_opNorm_le_of_le
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : SeminormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
{x : E} {a b : ℝ}, ‖f‖ ≤ a → ‖x‖ ≤ b → ‖f x‖ ≤ a * b- Defined in
- Mathlib.Analysis.Normed.Operator.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 165 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 · cited by 62,936
- Realstatement and proof · cited by 25,697
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement and proof · cited by 5,413
- ContinuousLinearMapstatement and proof · cited by 5,352
- LE.le.transproof · cited by 3,151
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- norm_nonnegproof · cited by 725
- RingHomIsometricstatement and proof · cited by 282
- mul_le_mulproof · cited by 144
Cited by4
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.le_of_opNorm_leproof · cited by 21
- ContinuousLinearMap.le_opNorm_of_leproof · cited by 5
- ContinuousLinearMap.le_of_opNorm₂_le_of_leproof · cited by 4
- Memℓp.holder_top_left_boundproof · cited by 1