Theorems · Definition · functional analysis

ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear

(𝕜 : Type u) →
  {ι : Type v} →
    (E : ι → Type wE) →
      (E₁ : ι → Type wE₁) →
        (G : Type wG) →
          [inst : NontriviallyNormedField 𝕜] →
            [inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] →
              [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)] →
                [inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)] →
                  [inst_4 : (i : ι) → NormedSpace 𝕜 (E₁ i)] →
                    [inst_5 : SeminormedAddCommGroup G] →
                      [inst_6 : NormedSpace 𝕜 G] →
                        [Fintype ι] →
                          ContinuousMultilinearMap 𝕜 (fun i => E i →L[𝕜] E₁ i)
                            (ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)

If f is a collection of continuous linear maps, then the construction ContinuousMultilinearMap.compContinuousLinearMap sending a continuous multilinear map g to g (f₁ ·, ..., fₙ ·) is continuous-linear in g and continuous-multilinear in f₁, ..., fₙ.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

9 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype