English
If A: F → G and B: E → F are linear maps between finite-dimensional inner product spaces, then the adjoint of their composition is the composition of their adjoints in reverse order: (A ∘ B)† = B† ∘ A†.
Русский
Если A: F → G и B: E → F — линейные отображения между конечномерными пространствами с имплицитной скалярной скалярной скалярной скалярной -> тогда сопряжение композиции равно композиции сопряжений в обратном порядке: (A ∘ B)† = B† ∘ A†.
LaTeX
$$$$(A \circ B)^{*} = B^{*} \circ A^{*}$$$$
Lean4
/-- The adjoint of the composition of two operators is the composition of the two adjoints
in reverse order. -/
@[simp]
theorem adjoint_comp (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F) : (A ∘ₗ B).adjoint = B.adjoint ∘ₗ A.adjoint :=
by
ext v
refine ext_inner_left 𝕜 fun w => ?_
simp only [adjoint_inner_right, LinearMap.coe_comp, Function.comp_apply]
