Theorems · Definition · functional analysis

PiTensorProduct.liftIsometry

{ι : Type u_1} →
  [inst : Fintype ι] →
    (𝕜 : Type u_2) →
      (E : ι → Type u_3) →
        [inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] →
          [inst_2 : NontriviallyNormedField 𝕜] →
            [inst_3 : (i : ι) → NormedSpace 𝕜 (E i)] →
              (F : Type u_4) →
                [inst_4 : SeminormedAddCommGroup F] →
                  [inst_5 : NormedSpace 𝕜 F] →
                    ContinuousMultilinearMap 𝕜 E F ≃ₗᵢ[𝕜] (PiTensorProduct 𝕜 fun i => E i) →L[𝕜] F

For a normed space F, we have constructed in PiTensorProduct.liftEquiv the canonical linear equivalence between ContinuousMultilinearMap 𝕜 E F and (⨂[𝕜] i, Eᵢ) →L[𝕜] F (induced by PiTensorProduct.lift). Here we give the upgrade of this equivalence to an isometric linear equivalence; in particular, it is a continuous linear equivalence.

Defined in

Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Cited by

4 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeSeminormedAddCommGroupNontriviallyNormedFieldNormedSpaceSeminormedAddCommGroupNormedSpace