Theorems · Theorem · functional analysis

PiTensorProduct.liftIsometry_comp_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)] {F : Type u_4}
  [inst_4 : SeminormedAddCommGroup F] [inst_5 : NormedSpace 𝕜 F] {E' : ι → Type u_5}
  [inst_6 : (i : ι) → SeminormedAddCommGroup (E' i)] [inst_7 : (i : ι) → NormedSpace 𝕜 (E' i)]
  (f : (i : ι) → E i →L[𝕜] E' i) (h : ContinuousMultilinearMap 𝕜 E' F),
  (PiTensorProduct.liftIsometry 𝕜 E' F) h ∘SL PiTensorProduct.mapL f =
    (PiTensorProduct.liftIsometry 𝕜 E F) (h.compContinuousLinearMap f)

Defined in

Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Cited by

0 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeSeminormedAddCommGroupNontriviallyNormedFieldNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpace