% custom commands go here %
 
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\renewcommand*{\O}{\mathcal{O}}
\renewcommand*{\epsilon}{\varepsilon}

\newcommand{\Rq}{\R^2}
\newcommand{\Qq}{\Q^2}

\newcommand*{\Rmn}{\R^{m\times n}}

 % round functions
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}

\newcommand{\lrarr}{\leftrightarrow}
\newcommand{\larr}{\leftarrow}
\newcommand{\rarr}{\rightarrow}
\newcommand{\Lrarr}{\Leftrightarrow}
\newcommand{\Larr}{\Leftarrow}
\newcommand{\Rarr}{\Rightarrow}

\newcommand*{\ex}{\emph{Beispiel: }}
\newcommand{\anm}{\emph{Anmerkung: }}
\newcommand{\prac}{\textbf{\textcolor{red}{Übung }}}

% TODO: improve usage of \def and \ther
\newcommand{\cdef}{\textbf{Definition: }}
\newcommand{\ther}{\textbf{Satz: }}
	
	
\newcommand{\linf}{\lim\limits_{n \rightarrow{} \infty}}

%running fraction with slash - requires math mode.
\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}

% Vector
\newcommand*\colvec[3][]{
	\begin{pmatrix}\ifx\relax#1\relax\else#1\\\fi#2\\#3\end{pmatrix}
}
\newcommand{\xyvec}{\colvec{x}{y}}

% Index command to show the key emphasized
\newcommand{\idx}[1]{{\emph{#1}\index{#1}}}

\newtheorem{satz}{Satz}[subsection]