Theorems · Theorem · functional analysis

isBoundedLinearMap_continuousMultilinearMap_comp_linear

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : SeminormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  {G : Type u_4} [inst_5 : SeminormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {ι : Type u_5} [inst_7 : Fintype ι]
  (g : G →L[𝕜] E), IsBoundedLinearMap 𝕜 fun f => f.compContinuousLinearMap fun x => g

Given a fixed continuous linear map g, associating to a continuous multilinear map f the continuous multilinear map f (g m₁, ..., g mₙ) is a bounded linear operation.

Defined in

Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype