Proof of Theorem IPVA

The proofs of the two parts are very similar, with the second one requiring just a bit more effort due to the conjugation that occurs. We will prove part 1 and you can prove part 2 (Exercise O.T10). \begin{align*} \innerproduct{\vect{u}+\vect{v}}{\vect{w}} &=\sum_{i=1}^{m}\conjugate{\vectorentry{\vect{u}+\vect{v}}{i}}\vectorentry{\vect{w}}{i} &&\knowl{./knowls/definition.IP.knowl}{\text{Definition IP}}\\ &=\sum_{i=1}^{m}\left(\conjugate{\vectorentry{\vect{u}}{i}+ \vectorentry{\vect{v}}{i}}\right)\vectorentry{\vect{w}}{i} &&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &=\sum_{i=1}^{m}\left(\conjugate{\vectorentry{\vect{u}}{i}}+ \conjugate{\vectorentry{\vect{v}}{i}}\right)\vectorentry{\vect{w}}{i} &&\knowl{./knowls/theorem.CCRA.knowl}{\text{Theorem CCRA}}\\ &=\sum_{i=1}^{m}\conjugate{\vectorentry{\vect{u}}{i}}\vectorentry{\vect{w}}{i} + \conjugate{\vectorentry{\vect{v}}{i}}\vectorentry{\vect{w}}{i} &&\knowl{./knowls/property.DCN.knowl}{\text{Property DCN}}\\ &=\sum_{i=1}^{m}\conjugate{\vectorentry{\vect{u}}{i}}\vectorentry{\vect{w}}{i} +\sum_{i=1}^{m}\conjugate{\vectorentry{\vect{v}}{i}}\vectorentry{\vect{w}}{i} &&\knowl{./knowls/property.CACN.knowl}{\text{Property CACN}}\\ &=\innerproduct{\vect{u}}{\vect{w}}+\innerproduct{\vect{v}}{\vect{w}}&&\knowl{./knowls/definition.IP.knowl}{\text{Definition IP}} \end{align*}