\section{Vectors and bases}
A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$
\section{Vectors and bases}
A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$
\label{eq:vec_sum}
\end{equation}
i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
\label{eq:vec_sum}
\end{equation}
i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
-Indeed, the very important identity representation by the outer product ($\cdot$) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.
+Indeed, the very important identity representation by the outer product ($\otimes$, see \ref{math_app:product}) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.