Theorems · Definition · functional analysis

ContinuousLinearMap.precomp

{𝕜₁ : Type u_1} →
  {𝕜₂ : Type u_2} →
    {𝕜₃ : Type u_3} →
      [inst : NormedField 𝕜₁] →
        [inst_1 : NormedField 𝕜₂] →
          [inst_2 : NormedField 𝕜₃] →
            {σ : 𝕜₁ →+* 𝕜₂} →
              {τ : 𝕜₂ →+* 𝕜₃} →
                {ρ : 𝕜₁ →+* 𝕜₃} →
                  [RingHomCompTriple σ τ ρ] →
                    {E : Type u_4} →
                      {F : Type u_5} →
                        (G : Type u_6) →
                          [inst_4 : AddCommGroup E] →
                            [inst_5 : Module 𝕜₁ E] →
                              [inst_6 : AddCommGroup F] →
                                [inst_7 : Module 𝕜₂ F] →
                                  [inst_8 : AddCommGroup G] →
                                    [inst_9 : Module 𝕜₃ G] →
                                      [inst_10 : TopologicalSpace E] →
                                        [inst_11 : TopologicalSpace F] →
                                          [inst_12 : TopologicalSpace G] →
                                            [inst_13 : IsTopologicalAddGroup G] →
                                              [inst_14 : ContinuousConstSMul 𝕜₃ G] →
                                                [RingHomSurjective σ] →
                                                  [RingHomIsometric σ] → (E →SL[σ] F) → (F →SL[τ] G) →L[𝕜₃] E →SL[ρ] G

Pre-composition by a fixed continuous linear map as a continuous linear map. Note that in non-normed space it is not always true that composition is continuous in both variables, so we have to fix one of them.

Defined in

Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Cited by

13 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldNormedFieldNormedFieldRingHomCompTripleAddCommGroupModuleAddCommGroupModuleAddCommGroupModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceIsTopologicalAddGroupContinuousConstSMulRingHomSurjectiveRingHomIsometric