Theorems · Theorem · functional analysis

TensorProduct.adjoint_mapL

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {H : Type u_5} [inst : RCLike 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace 𝕜 F] [inst_5 : NormedAddCommGroup G] [inst_6 : InnerProductSpace 𝕜 G]
  [inst_7 : NormedAddCommGroup H] [inst_8 : InnerProductSpace 𝕜 H] [inst_9 : CompleteSpace E]
  [inst_10 : CompleteSpace G] [inst_11 : CompleteSpace (TensorProduct 𝕜 E G)] [inst_12 : CompleteSpace F]
  [inst_13 : CompleteSpace H] [inst_14 : CompleteSpace (TensorProduct 𝕜 F H)] (f : E →L[𝕜] F) (g : G →L[𝕜] H),
  ContinuousLinearMap.adjoint (TensorProduct.mapL f g) =
    TensorProduct.mapL (ContinuousLinearMap.adjoint f) (ContinuousLinearMap.adjoint g)

Defined in

Mathlib.Analysis.InnerProductSpace.TensorProduct

Cited by

2 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceCompleteSpaceCompleteSpaceCompleteSpaceCompleteSpaceCompleteSpaceCompleteSpace