Theorems · Definition · functional analysis

ContinuousMultilinearMap.compContinuousLinearMapL

{𝕜 : Type u_1} →
  {ι : Type u_2} →
    {E : ι → Type u_3} →
      {F : Type u_4} →
        [inst : NormedField 𝕜] →
          [inst_1 : (i : ι) → TopologicalSpace (E i)] →
            [inst_2 : (i : ι) → AddCommGroup (E i)] →
              [inst_3 : (i : ι) → Module 𝕜 (E i)] →
                [inst_4 : AddCommGroup F] →
                  [inst_5 : Module 𝕜 F] →
                    [inst_6 : TopologicalSpace F] →
                      [inst_7 : IsTopologicalAddGroup F] →
                        {E₁ : ι → Type u_5} →
                          [inst_8 : (i : ι) → TopologicalSpace (E₁ i)] →
                            [inst_9 : ContinuousConstSMul 𝕜 F] →
                              [inst_10 : (i : ι) → AddCommGroup (E₁ i)] →
                                [inst_11 : (i : ι) → Module 𝕜 (E₁ i)] →
                                  ((i : ι) → E i →L[𝕜] E₁ i) →
                                    ContinuousMultilinearMap 𝕜 E₁ F →L[𝕜] ContinuousMultilinearMap 𝕜 E F

ContinuousMultilinearMap.compContinuousLinearMap as a bundled continuous linear map. Given a family of continuous linear maps f : Π i, E i →L[𝕜] E₁ i, this function returns a continuous linear maps between the spaces of continuous multilinear maps on Π i, E₁ i and on Π i, E i. The map sends g to the map given by v ↦ g (fun i ↦ f i (v i)). Actually, the map is multilinear in f, see ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear. For a version fixing g and varying f, see compContinuousLinearMapLRight.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Topology

Cited by

13 results in Mathlib

Foundations

Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldTopologicalSpaceAddCommGroupModuleAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupTopologicalSpaceContinuousConstSMulAddCommGroupModule