\begin{definition}[Definition:Adjoint Linear Transformation]
Let $\HH$ and $\KK$ be Hilbert spaces.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.
By Existence and Uniqueness of Adjoint, there exists a unique bounded linear transformation $A^* \in \map \BB {\KK, \HH}$ such that:
:$\forall h \in \HH, k \in \KK: {\innerprod {\map A h} k}_\KK = {\innerprod h {\map {A^*} k} }_\HH$
where $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ are inner products on $\HH$ and $\KK$ respectively.
$A^*$ is called the '''adjoint''' of $A$.
The operation of assigning $A^*$ to $A$ may be referred to as '''adjoining'''.
\end{definition}