\usepackage[utf8]{inputenc}
\usepackage[danish]{babel}
\usepackage{natbib}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[margin=0.8in]{geometry}
\usepackage{tikz}

% often used sets: N, Z, Q, R, C, F
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}

\newcommand{\fat}[1]{\mathbf{#1}}
\newcommand{\curly}[1]{\{#1\}}
\newcommand{\para}[1]{\left(#1\right)}
\newcommand{\makeset}[2]{\curly{#1 \mid #2}}
\newcommand{\openint}[2]{\mathopen]#1, #2\mathclose[}
\newcommand{\closedint}[2]{\mathopen[#1, #2\mathclose]}
\newcommand{\fracp}[2]{\left(\frac{#1}{#2}\right)}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\ip}[2]{\langle #1, #2\rangle}
\newcommand{\paraip}[2]{\left\langle #1, #2\right\rangle}

% \vrep{x}{V}
% \mrep{W}{L}{V}
\newcommand{\vrep}[2]{[#1]_{\mathcal{#2}}}
\newcommand{\mrep}[3]{{}_{\mathcal{#1}}[#2]_{\mathcal{#3}}}
\newcommand{\pdif}[2]{\frac{\partial #1}{\partial #2}}

\newcommand{\seq}[2]{\curly{#1}_{#2 = 1}^{\infty}}
\newcommand{\series}[2]{\sum_{#1}^{\infty} #2}

\newcommand{\funcdef}[5]{
    \begin{align*}
        #1 \colon #2 &\to #3 \\
        #4 &\mapsto #5
    \end{align*}
}

\makeatletter                                      % adds vertical lines to matrices
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

% Gør det muligt at lave 'cases' environmentet vendt om
\newenvironment{rcases}
  {\left.\begin{aligned}}
  {\end{aligned}\right\rbrace}

% Checkmark, fra: https://tex.stackexchange.com/a/132790
\def\checkmark{\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;}