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diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index e6e935e..edee2ff 100644
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -1,6 +1,8 @@
 \chapter{Mathematical tools}
 
-\section{Vector space}
+\section{Vector algebra}
+
+\subsection{Vector space}
 \label{math_app:vector_space}
 
 \begin{definition}
@@ -32,6 +34,19 @@ Due to the additive abelian group, the following properties are additionally val
 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}