Theorems · Definition · functional analysis

MultilinearMap.mkContinuousLinear

{𝕜 : Type u} →
  {ι : Type v} →
    {E : ι → Type wE} →
      {G : Type wG} →
        {G' : Type wG'} →
          [inst : NontriviallyNormedField 𝕜] →
            [inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] →
              [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)] →
                [inst_3 : SeminormedAddCommGroup G] →
                  [inst_4 : NormedSpace 𝕜 G] →
                    [inst_5 : SeminormedAddCommGroup G'] →
                      [inst_6 : NormedSpace 𝕜 G'] →
                        [inst_7 : Fintype ι] →
                          (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') →
                            (C : ℝ) →
                              (∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) →
                                G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'

Given a map f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' and an estimate H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖, construct a continuous linear map from G to ContinuousMultilinearMap 𝕜 E G'. In order to lift, e.g., a map f : (MultilinearMap 𝕜 E G) →ₗ[𝕜] MultilinearMap 𝕜 E' G' to a map (ContinuousMultilinearMap 𝕜 E G) →L[𝕜] ContinuousMultilinearMap 𝕜 E' G', one can apply this construction to f.comp ContinuousMultilinearMap.toMultilinearMapLinear which is a linear map from ContinuousMultilinearMap 𝕜 E G to MultilinearMap 𝕜 E' G'.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

2 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype