Mathlib Map

Theorems Β· Definition Β· functional analysis

LinearEquiv.extend

{π•œ : Type u_1} β†’
  {π•œβ‚‚ : Type u_2} β†’
    {E : Type u_3} β†’
      {Eβ‚— : Type u_4} β†’
        {F : Type u_5} β†’
          {Fβ‚— : Type u_6} β†’
            [inst : NormedDivisionRing π•œ] β†’
              [inst_1 : NormedDivisionRing π•œβ‚‚] β†’
                [inst_2 : AddCommGroup E] β†’
                  [inst_3 : NormedAddCommGroup Eβ‚—] β†’
                    [inst_4 : AddCommGroup F] β†’
                      [inst_5 : NormedAddCommGroup Fβ‚—] β†’
                        [inst_6 : Module π•œ E] β†’
                          [inst_7 : Module π•œ Eβ‚—] β†’
                            [IsBoundedSMul π•œ Eβ‚—] β†’
                              [inst_9 : Module π•œβ‚‚ F] β†’
                                [inst_10 : Module π•œβ‚‚ Fβ‚—] β†’
                                  [IsBoundedSMul π•œβ‚‚ Fβ‚—] β†’
                                    [CompleteSpace Eβ‚—] β†’
                                      [CompleteSpace Fβ‚—] β†’
                                        {σ₁₂ : π•œ β†’+* π•œβ‚‚} β†’
                                          {σ₂₁ : π•œβ‚‚ β†’+* π•œ} β†’
                                            [inst_14 : RingHomInvPair σ₁₂ σ₂₁] β†’
                                              [inst_15 : RingHomInvPair σ₂₁ σ₁₂] β†’
                                                (f : E ≃ₛₗ[σ₁₂] F) β†’
                                                  (e₁ : E β†’β‚—[π•œ] Eβ‚—) β†’
                                                    (eβ‚‚ : F β†’β‚—[π•œβ‚‚] Fβ‚—) β†’
                                                      DenseRange ⇑e₁ β†’
                                                        (βˆƒ C, βˆ€ (x : E), β€–eβ‚‚ (f x)β€– ≀ C * β€–e₁ xβ€–) β†’
                                                          DenseRange ⇑eβ‚‚ β†’
                                                            (βˆƒ C, βˆ€ (x : F), β€–e₁ (f.symm x)β€– ≀ C * β€–eβ‚‚ xβ€–) β†’
                                                              Eβ‚— ≃SL[σ₁₂] Fβ‚—

Extension of a linear equivalence f : E ≃ₛₗ[σ₁₂] F to a continuous linear equivalence Eβ‚— ≃SL[σ₁₂] Fβ‚—, where E and F are normed spaces and Eβ‚— and Fβ‚— are Banach spaces, using dense maps e₁ : E β†’β‚—[π•œβ‚] Eβ‚— and eβ‚‚ : F β†’β‚—[π•œβ‚‚] Fβ‚‚ together with bounds β€–eβ‚‚ (f x)β€– ≀ C * β€–e₁ xβ€– for all x : E and β€–e₁ (f.symm x)β€– ≀ C * β€–eβ‚‚ xβ€– for all x : F.

Defined in
Mathlib.Analysis.Normed.Operator.Extend
Cited by
7 results in Mathlib
Foundations
Depth 167 from the axioms Β· uses propext, Classical.choice, Quot.sound
Assumes
NormedDivisionRingNormedDivisionRingAddCommGroupNormedAddCommGroupAddCommGroupNormedAddCommGroupModuleModuleIsBoundedSMulModuleModuleIsBoundedSMulCompleteSpaceCompleteSpaceRingHomInvPairRingHomInvPair
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

LinearEquiv.extendOfIsometry Β· cited by 4LinearEquiv.extendOfIsome…LinearEquiv.extend.congr_simp Β· cited by 0extend.congr_simpLinearEquiv.extend_apply Β· cited by 0LinearEquiv.extend_applyLinearEquiv.extend_eq Β· cited by 0LinearEquiv.extend_eqLinearEquiv.extend_symm_apply Β· cited by 0LinearEquiv.extend_symm_a…LinearEquiv.extend_symm_eq Β· cited by 0LinearEquiv.extend_symm_eqLinearEquiv.norm_extend_le Β· cited by 0LinearEquiv.norm_extend_leLinearEquiv.norm_extend_symm_le Β· cited by 0LinearEquiv.norm_extend_s…DFunLike.coe Β· cited by 62936DFunLike.coeReal Β· cited by 25697RealModule Β· cited by 20661ModuleRingHom.id Β· cited by 18349RingHom.idNormedAddCommGroup Β· cited by 15752NormedAddCommGroupAddCommGroup Β· cited by 12871AddCommGroupLinearMap Β· cited by 10215LinearMapRingHom Β· cited by 10189RingHomNorm.norm Β· cited by 5413Norm.normContinuousLinearMap Β· cited by 5352ContinuousLinearMapLinearEquiv Β· cited by 3317LinearEquivCompleteSpace Β· cited by 2532CompleteSpaceLinearMap.comp Β· cited by 1642LinearMap.compLinearEquiv.symm Β· cited by 1461LinearEquiv.symmLinearEquiv.toLinearMap Β· cited by 1171LinearEquiv.toLinearMapLinearEquiv.extendCITED BYCITES

Cites22

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Cited by8

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