Theorems · Theorem · functional analysis

TensorProduct.adjoint_map

∀ {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 : FiniteDimensional 𝕜 E]
  [inst_10 : FiniteDimensional 𝕜 F] [inst_11 : FiniteDimensional 𝕜 G] [inst_12 : FiniteDimensional 𝕜 H] (f : E →ₗ[𝕜] F)
  (g : G →ₗ[𝕜] H),
  LinearMap.adjoint (TensorProduct.map f g) = TensorProduct.map (LinearMap.adjoint f) (LinearMap.adjoint g)

Defined in

Mathlib.Analysis.InnerProductSpace.TensorProduct

Cited by

2 results in Mathlib

Foundations

Depth 246 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceFiniteDimensionalFiniteDimensionalFiniteDimensionalFiniteDimensional