Proof of Theorem MMIP

\begin{align*} \innerproduct{\vect{u}}{\vect{v}} &=\sum_{k=1}^{m}\conjugate{\vectorentry{\vect{u}}{k}}\vectorentry{\vect{v}}{k} &&\knowl{./knowls/definition.IP.knowl}{\text{Definition IP}}\\ &=\sum_{k=1}^{m}\conjugate{\matrixentry{\vect{u}}{k1}}\matrixentry{\vect{v}}{k1} &&\text{Column vectors as matrices}\\ &=\sum_{k=1}^{m}\matrixentry{\conjugate{\vect{u}}}{k1}\matrixentry{\vect{v}}{k1} &&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\sum_{k=1}^{m}\matrixentry{\transpose{\conjugate{\vect{u}}}}{1k}\matrixentry{\vect{v}}{k1} &&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\matrixentry{\transpose{\conjugate{\vect{u}}}\vect{v}}{11} &&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}}\\ \end{align*}

To finish we just blur the distinction between a $1\times 1$ matrix ($\transpose{\conjugate{\vect{u}}}\vect{v}$) and its lone entry.