Theorems · Theorem · functional analysis
ContinuousLinearMap.lipschitz
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NontriviallyNormedField 𝕜₂] [inst_2 : SeminormedAddCommGroup E] [inst_3 : SeminormedAddCommGroup F]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F), LipschitzWith ‖f‖₊ ⇑fcontinuous linear maps are Lipschitz continuous.
- Defined in
- Mathlib.Analysis.Normed.Operator.NNNorm
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- ContinuousLinearMapstatement and proof · cited by 5,352
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- NNNorm.nnnormstatement and proof · cited by 952
- LipschitzWithstatement · cited by 316
- RingHomIsometricstatement and proof · cited by 282
- ContinuousLinearMap.le_opNNNormproof · cited by 6
- AddMonoidHomClass.lipschitz_of_bound_nnnormproof · cited by 1
Cited by16
Results whose statement or proof uses this declaration.
- MeasureTheory.MemLp.continuousLinearMap_compproof · cited by 5
- ContinuousLinearEquiv.lipschitzproof · cited by 5
- LipschitzWith.ae_lineDifferentiableAtproof · cited by 3
- ContinuousLinearMap.integral_comp_comm'proof · cited by 2
- RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrictproof · cited by 1
- ApproximatesLinearOn.lipschitzproof · cited by 1
- MeasureTheory.completeSpace_of_completeSpace_Lpproof · cited by 1
- LipschitzWith.hasFDerivAt_of_hasLineDerivAt_of_closureproof · cited by 1
- ContinuousLinearMap.isCompact_closure_image_coe_of_boundedproof · cited by 1
- SeparatingDual.completeSpace_of_completeSpace_continuousLinearMapproof · cited by 1
- SeparatingDual.completeSpace_of_completeSpace_continuousMultilinearMapproof · cited by 1
- Complex.dist_le_mul_div_pow_of_mapsTo_ball_of_isLittleOproof · cited by 1