Given the vectors $\vect{u},\,\vect{v}\in\complex{m}$ the inner product of $\vect{u}$ and $\vect{v}$ is the scalar quantity in $\complex{\null}$, \begin{equation*} \innerproduct{\vect{u}}{\vect{v}}= \conjugate{\vectorentry{\vect{u}}{1}}\vectorentry{\vect{v}}{1}+ \conjugate{\vectorentry{\vect{u}}{2}}\vectorentry{\vect{v}}{2}+ \conjugate{\vectorentry{\vect{u}}{3}}\vectorentry{\vect{v}}{3}+ \cdots+ \conjugate{\vectorentry{\vect{u}}{m}}\vectorentry{\vect{v}}{m} = \sum_{i=1}^{m}\conjugate{\vectorentry{\vect{u}}{i}}\vectorentry{\vect{v}}{i} \end{equation*}