Adjoint of Composition of Linear Transformations is Composition of Adjoints

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $\HH$, $\KK$ and $\LL$ be a Hilbert spaces over $\mathbb F$.

Let $A : \KK \to \LL$ and $B : \HH \to \KK$ be bounded linear transformations.

Then:
 * $\paren {A B}^* = B^* A^*$

where $^*$ denotes adjoining.

Proof
Let ${\innerprod \cdot \cdot}_\HH$, ${\innerprod \cdot \cdot}_\KK$ and ${\innerprod \cdot \cdot}_\LL$ denote inner products over $\HH$, $\KK$ and $\LL$ respectively.

Let $h \in \HH$ and $l \in \LL$.

Then:

We also have, by the definition of the adjoint:


 * $\innerprod {\map {\paren {A B} } h} l_\LL = \innerprod h {\map {\paren {A B}^*} l}_\LL$

So, by the uniqueness part of Existence and Uniqueness of Adjoint:
 * $\paren {A B}^* = B^* A^*$