Theorems · Theorem · functional analysis

LinearMap.adjoint_comp

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedAddCommGroup G] [inst_4 : InnerProductSpace 𝕜 E]
  [inst_5 : InnerProductSpace 𝕜 F] [inst_6 : InnerProductSpace 𝕜 G] [inst_7 : FiniteDimensional 𝕜 E]
  [inst_8 : FiniteDimensional 𝕜 F] [inst_9 : FiniteDimensional 𝕜 G] (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F),
  LinearMap.adjoint (A ∘ₗ B) = LinearMap.adjoint B ∘ₗ LinearMap.adjoint A

The adjoint of the composition of two operators is the composition of the two adjoints in reverse order.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

1 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceInnerProductSpaceFiniteDimensionalFiniteDimensionalFiniteDimensional