\part{Mathematical foundations}
-Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
-
\chapter{Linear algebra}
+Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
+
\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.
-The scalar product for an $N$-dimensional real vector space is defined as
+The scalar product in an $N$-dimensional Euclidean vector space is defined as
\begin{equation}
(\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
\label{eq:vec_sp}
\end{equation}
-which enables to define a norm
+which satisfies the properties of an inner product (see \ref{math_app:product}) and enables to define a norm
\begin{equation}
||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
\end{equation}
\begin{equation}
\label{eq:complete}
\vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow
-\sum_i\vec{e}_i\cdot \vec{e}_i=\vec{1}
+\sum_i\vec{e}_i\otimes \vec{e}_i=\vec{1}
\end{equation}
if the basis is complete.
-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.
\section{Operators, matrices and determinants}