\begin{equation}
||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
\end{equation}
-that just corresponds to the length of vector \vec{a}.
+that just corresponds to the length of vector $\vec{a}$.
Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
\begin{equation}
(\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=