Theorems · Definition · functional analysis

ContinuousLinearMap.bilinearComp

{𝕜 : Type u_1} →
  {𝕜₂ : Type u_2} →
    {𝕜₃ : Type u_3} →
      {E : Type u_4} →
        {F : Type u_6} →
          {G : Type u_8} →
            [inst : SeminormedAddCommGroup E] →
              [inst_1 : SeminormedAddCommGroup F] →
                [inst_2 : SeminormedAddCommGroup G] →
                  [inst_3 : NontriviallyNormedField 𝕜] →
                    [inst_4 : NontriviallyNormedField 𝕜₂] →
                      [inst_5 : NontriviallyNormedField 𝕜₃] →
                        [inst_6 : NormedSpace 𝕜 E] →
                          [inst_7 : NormedSpace 𝕜₂ F] →
                            [inst_8 : NormedSpace 𝕜₃ G] →
                              {σ₂₃ : 𝕜₂ →+* 𝕜₃} →
                                {σ₁₃ : 𝕜 →+* 𝕜₃} →
                                  {E' : Type u_11} →
                                    {F' : Type u_12} →
                                      [inst_9 : SeminormedAddCommGroup E'] →
                                        [inst_10 : SeminormedAddCommGroup F'] →
                                          {𝕜₁' : Type u_13} →
                                            {𝕜₂' : Type u_14} →
                                              [inst_11 : NontriviallyNormedField 𝕜₁'] →
                                                [inst_12 : NontriviallyNormedField 𝕜₂'] →
                                                  [inst_13 : NormedSpace 𝕜₁' E'] →
                                                    [inst_14 : NormedSpace 𝕜₂' F'] →
                                                      {σ₁' : 𝕜₁' →+* 𝕜} →
                                                        {σ₁₃' : 𝕜₁' →+* 𝕜₃} →
                                                          {σ₂' : 𝕜₂' →+* 𝕜₂} →
                                                            {σ₂₃' : 𝕜₂' →+* 𝕜₃} →
                                                              [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] →
                                                                [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] →
                                                                  [RingHomIsometric σ₂₃] →
                                                                    [RingHomIsometric σ₁₃'] →
                                                                      [RingHomIsometric σ₂₃'] →
                                                                        (E →SL[σ₁₃] F →SL[σ₂₃] G) →
                                                                          (E' →SL[σ₁'] E) →
                                                                            (F' →SL[σ₂'] F) →
                                                                              E' →SL[σ₁₃'] F' →SL[σ₂₃'] G

Compose a bilinear map E →SL[σ₁₃] F →SL[σ₂₃] G with two linear maps E' →SL[σ₁'] E and F' →SL[σ₂'] F.

Defined in

Mathlib.Analysis.Normed.Operator.Bilinear

Cited by

11 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceSeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomCompTripleRingHomCompTripleRingHomIsometricRingHomIsometricRingHomIsometric