Sejam $u$, $v$ e $w$ vetores do $\mathbb{C}^n$, mostrar que $u \cdot (v + w) = u \cdot v + u \cdot w$.

Sejam $u_i$, $v_i$ e $w_i$ as $i$-ésimas coordenadas de $u$, $v$ e $w$ respectivamente.

A $i$-ésima parcela de $u \cdot (v + w)$ será $u_i \cdot \overline{(v_i + w_i)} = u_i\overline{v_i} + u_i\overline{w_i}$. Logo:

$u \cdot (v + w) = \displaystyle\sum_{i=1}^n (u_i\overline{v_i} + u_i\overline{w_i}) = \displaystyle\sum_{i=1}^n u_i\overline{v_i} + \displaystyle\sum_{i=1}^n u_i\overline{w_i} = u \cdot v + u \cdot w$.

Quod Erat Demonstrandum.