\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{polski}
\usepackage{amsmath}
\usepackage{amsthm}
\begin{document}

\begin{enumerate}
\theoremstyle{definition}
\newtheorem{thm}{Theorem}

\newtheorem{df}{Definition}

\item Properly typeset the following command and properly refere to it in the text
\begin{align*}
&\sum_{i_1,\dots,i_m}  a_{i_1,\dots,i_m}^{2m}{m+1} {\frac{m+1}{2m}} \leq \\
C \sup\Big\{ |&\sum_{i_1,\dots, i_m} a_{i_1,\dots,i_m} x^1_{i_1}\dots x^m_{i_m}|: \|(x_i^k)_{i=1}^n \|_\infty\leq1,\ 1\leq k\leq m\Big\},
\end{align*}
\item Properly typset the expression: $\operatorname{Re} z$.

\item Properly typeset indexes in the following sum:
\[
f(x)=\sum_{\substack{n=0\\ k=2}}^\infty a_n^k
\]
\item Properly typeset the following theorem
\begin{thm}[Cauchy--Hadamard]
The radius of convergence $R$ of the power series
\[
\sum_{n=0}^\infty a_n(z-z_0)^n,\ \ \ \ \ |z-z_0|<R
\]
can by calculated via the following formula
\[
\frac{1}{R}=\limsup_{n\to\infty} \sqrt[n]{|a_n|}.
\]
\end{thm}
\begin{df}[Prime numbers]
A number is called prime, if it is not compound. 

\end{df}

\item Typeset the follwing matrix (display-style):
\[
\begin{pmatrix}
         a_{11} & a_{12}\\
         a_{21} & a_{22}
\end{pmatrix}
\]
and in the text $\left\{\begin{smallmatrix}
                            a_{11} & a_{12}\\
                            a_{21} & a_{22}
                          \end{smallmatrix}
     \right\}$
     Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do
     eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut
     enim ad minim veniam, quis nostrud exercitation ullamco laboris
     nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in
     reprehenderit in voluptate velit esse cillum dolore eu fugiat
     nulla pariatur. Excepteur sint occaecat cupidatat non proident,
     sunt in culpa qui officia deserunt mollit anim id est laborum.

\end{enumerate}
\end{document}