If we consider the vectors $\vect{u},\,\vect{v}\in\complex{m}$ as $m\times 1$ matrices then \begin{align*} \innerproduct{\vect{u}}{\vect{v}} &=\transpose{\conjugate{\vect{u}}}\vect{v} =\adjoint{\vect{u}}\vect{v} \end{align*}