\chapter{Mathematical tools}
-\section{Vector space}
+\section{Vector algebra}
+
+\subsection{Vector space}
\label{math_app:vector_space}
\begin{definition}
The addition of two vectors is called vector addition.
\end{remark}
+\subsection{Dual space}
+
+\subsection{Inner and outer product}
+\label{math_app:product}
+
+\begin{definition}
+The inner product ...
+\end{definition}
+
+\begin{definition}
+If $\vec{u}\in U$ and $\vec{v}\in V$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$, the outer product of $\vec{u}$ and $\vec{v}$ is defined as the tensor product ...
+\end{definition}
+
\section{Spherical coordinates}
\section{Fourier integrals}