Theorems · Definition · functional analysis

PiTensorProduct.mapL

{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)] →
              {E' : ι → Type u_5} →
                [inst_4 : (i : ι) → SeminormedAddCommGroup (E' i)] →
                  [inst_5 : (i : ι) → NormedSpace 𝕜 (E' i)] →
                    ((i : ι) → E i →L[𝕜] E' i) → (PiTensorProduct 𝕜 fun i => E i) →L[𝕜] PiTensorProduct 𝕜 fun i => E' i

Let Eᵢ and E'ᵢ be two families of normed 𝕜-vector spaces. Let f be a family of continuous 𝕜-linear maps between Eᵢ and E'ᵢ, i.e. f : Πᵢ Eᵢ →L[𝕜] E'ᵢ, then there is an induced continuous linear map ⨂ᵢ Eᵢ → ⨂ᵢ E'ᵢ by ⨂ aᵢ ↦ ⨂ fᵢ aᵢ.

Defined in

Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Cited by

13 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeSeminormedAddCommGroupNontriviallyNormedFieldNormedSpaceSeminormedAddCommGroupNormedSpace