Theorems · Definition · functional analysis

ContinuousLinearEquiv.continuousMultilinearMapCongrLeft

{𝕜 : Type u_1} →
  {ι : Type u_2} →
    {E : ι → Type u_3} →
      {E₁ : ι → Type u_4} →
        (F : Type u_5) →
          [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 : (i : ι) → TopologicalSpace (E₁ i)] →
                    [inst_5 : (i : ι) → AddCommGroup (E₁ i)] →
                      [inst_6 : (i : ι) → Module 𝕜 (E₁ i)] →
                        [inst_7 : AddCommGroup F] →
                          [inst_8 : Module 𝕜 F] →
                            [inst_9 : TopologicalSpace F] →
                              [inst_10 : IsTopologicalAddGroup F] →
                                [inst_11 : ContinuousConstSMul 𝕜 F] →
                                  ((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 equiv. Given a family of continuous linear equivalences f : Π i, E i ≃L[𝕜] E₁ i, this function returns a continuous linear equivalence between the space of continuous multilinear maps with domain Π i, E i and codomain F and the space of multilinear maps with domain Π i, E₁ i and the same codomain, by composing the multilinear maps with f.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Topology

Cited by

3 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldTopologicalSpaceAddCommGroupModuleTopologicalSpaceAddCommGroupModuleAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMul