Theorems · Definition · functional analysis

PointwiseConvergenceCLM.piEquivL

(𝕜 : Type u_3) →
  [inst : NormedField 𝕜] →
    (E : Type u_7) →
      [inst_1 : AddCommGroup E] →
        [inst_2 : TopologicalSpace E] →
          [inst_3 : Module 𝕜 E] →
            {ι : Type u_11} →
              (F : ι → Type u_12) →
                [inst_4 : (i : ι) → AddCommGroup (F i)] →
                  [inst_5 : (i : ι) → Module 𝕜 (F i)] →
                    [inst_6 : (i : ι) → TopologicalSpace (F i)] →
                      [inst_7 : ∀ (i : ι), IsTopologicalAddGroup (F i)] →
                        [inst_8 : ∀ (i : ι), ContinuousConstSMul 𝕜 (F i)] →
                          ((i : ι) → E →Lₚₜ[𝕜] F i) ≃L[𝕜] E →Lₚₜ[𝕜] (i : ι) → F i

ContinuousLinearMap.pi, upgraded to a continuous linear equivalence between Π i, E →Lₚₜ[𝕜] F i and E →Lₚₜ[𝕜] Π i, F i.

Defined in

Mathlib.Topology.Algebra.Module.Spaces.PointwiseConvergenceCLM

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupTopologicalSpaceModuleAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMul