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\begin{document}
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Formulace Pythagorovy věty v analytické geometrii vypadá takto: $$\overrightarrow{AC}\bot\overrightarrow{BC}\Leftrightarrow |\overrightarrow{AC}|^2+|\overrightarrow{BC}|^2=|\overrightarrow{AB}|^2.$$
Ve středoškolské formulaci, reprezentované vzorcem \[a^2+b^2=c^2,\] znamenají $a$ a $b$ délky odvěsen a $c$ délku přepony. Protože například platí \(3^2+4^2=5^2\), je trojúhelník se stranami délek $3$, $-4$ a $5$ pravoúhlý.

\oddel
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\mathcal{MATHCAL}\\
\mathbb{MATHBB} \\%amsfonts
a>b \text{ a } b>c\text{, a tedy } a>c$\bigskip

$sin x$, $\sin x$ $\sin(x)$\\
$\tan x$, $\tg x$\\
špatně: $e^x$, $1+2i$, $\int x^2 dx$\\
\newcommand{\e}{\mathrm{e}}
správně: $\mathrm{e}^x$ $\e^x$, $1+2\mathrm{i}$, $\int x^2\,\mathrm{d}x$

\oddel
$\displaystyle\frac{1}{2}$, $\textstyle\frac{1}{2}$, $\scriptstyle\frac{1}{2}$, $\scriptscriptstyle\frac{1}{2}$

\oddel
$x^-1$, $x^{-1}$, $x^{2^n}$, $x_1$, $x_1^2$, $x^2_1$, $x_1{}^2$\\
$\sin_x,\ \lim_{x\to 1}$
\[\sin_x,\ \lim_{x\to 1}\]
$\sideset{_1^2}{_3^4}\sum$, $\overset{5}{X}$, $\underset{6}{X}$

\oddel
$x/y\text{, }\frac{x}{y}\text{, }\frac{1+\frac{x}{y}}{1-\frac{x}{y}}$
\[x/y\text{, }\frac{x}{y}\text{, }\frac{1+\frac{x}{y}}{1-\frac{x}{y}}\]
$\dfrac{x}{y}\text{, }\dfrac{1+\frac{x}{y}}{1-\frac{x}{y}}$\\
$\binom{n}{k}$

\[\frac{1}{2^{n^2}} \frac{1}{2\strut^{n^2}}\]

\oddel
$\sqrt[3]{x^2-1}$, $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}$

\oddel
$2.3^n$, $2\cdot 3^n$\\
$a_1,\ldots,a_n$, $a_1+\cdots+a_n$\\
$\vdots$, $\ddots$ -- viz matice

\oddel
$\pi\doteq 3,1415926535$\\
$\pi\doteq 3{,}1415926535$

\oddel
$\alpha\beta\gamma\delta\epsilon\varepsilon\phi\varphi$\\
$\Rightarrow\Leftarrow\Leftrightarrow\nearrow\searrow\smile\frown$\\
$\leq\geq\neq\ll\gg\subset\in \not\in \simeq\equiv$\\
$\pm\mp\times\circ\cap\cup\vee\wedge\setminus\smallsetminus$\\
$\emptyset\forall\exists\neg\partial\infty$\\
$\sum\prod\int\bigcap\bigcup\bigvee\bigwedge$\\
\[\sum\prod\int\bigcap\bigcup\bigvee\bigwedge\]
$\sum_{n=1}^{\infty} \frac{1}{2^n}$\\
\[\sum_{n=1}^{\infty} \frac{1}{2^n}\]
$\sum\limits_{n=1}^{\infty} \frac{1}{2^n}$\\
\[\sum\nolimits_{n=1}^{\infty} \frac{1}{2^n}\]
\[\int\int_A,\ \iint_A,\ \iiint_A,\ \iiiint_A,\ \idotsint_A\]

\oddel
$\dot{a}$, $\ddot{a}$, $\dddot{a}$, $\ddddot{a}$, $\bar{a}$, $\vec{a}$, $\vec{i}$, $\vec{\imath}$, $\vec{\jmath}$
\bigskip

$\bar{AB}$, $\vec{AB}$, $\overline{AB}$, $\underline{AB}$, $\overrightarrow{AB}$, $\overleftarrow{AB}$, $\overleftrightarrow{AB}$, $\underrightarrow{AB}$, $\underleftarrow{AB}$, $\underleftrightarrow{AB}$, $\overbrace{ABCD}^a$, $\underbrace{ABCD}_b$\\
$\overline{\overline{z}^2+1}$
\[\overbrace{a+\underbrace{b+\cdots+y}_{\alpha_i}+z}^{\text{abeceda}}\]

\oddel
$A\xleftarrow[T]{n+\mu-1}B\xrightarrow[T]{n+\mu-1}C$, $A\stackrel{V1}{=}B$

\oddel
$()[] \{\} | \| \lfloor\rfloor \lceil\rceil \langle0,1\rangle\ <a,b>$
\[\left(1-\sqrt[3]{x\left(\frac{x^2}{\left|x^3\right|}-1\right)}\right)^2\]

\oddel
Mezery:\\
$xx$ bez mezery\\
$x\,x$ malá mezera, $3/18$ čtverčíku\\
$x\:x$ střední mezera, $4/18$ čtverčíku\\
$x\;x$ velká mezera, $5/18$ čtverčíku\\
$x\!x$ záporná malá mezera, $-3/18$ čtverčíku\\
$x\mspace{-5mu}x$ obecná mezera
\bigskip

$\sqrt{2}x$, $\sqrt{2}\,x$\\
$\dfrac{1}{2}x$, $\dfrac{1}{2}\,x$

\oddel
\begin{multline*}
e_n=\sum_{k=0}^n c_{k,n}\frac{x^k}{k!} \geq \sum_{k=0}^m c_{k,n}\frac{x^k}{k!}+c_{m+1,n}\frac{x^{m+1}}{(m+1)!} \geq \\
\left(\sum_{k=0}^m \frac{x^k}{k!}-\frac{1}{2}\frac{x^{m+1}}{m+1!}\right)+\left(\frac{x^{m+1}}{(m+1)!}-\frac{1}{2}\frac{x^{m+1}}{(m+1)!}\right)=s_m
\text{\quad pro s.v. }n
\end{multline*}

\oddel
\[e_n=\sum_{k=0}^n \binom{n}{k} \left(\frac{x}{n}\right)^k=\sum_{k=0}^n \frac{n(n-1)\dots(n-k+1)}{n^k}\cdot\frac{x^k}{k!}=\sum_{k=0}^n c_{k,n}\frac{x^k}{k!}\]
\[\label{defexp1}\forall m\in\mathbb{N}_0 \forall n\geq m: \sum_{k=0}^m c_{k,n}\frac{x^k}{k!} \leq \sum_{k=0}^m \frac{x^k}{k!}=s_m\]
\[\frac{x^{m+1}}{(m+1)!}-c_{m+1,n}\frac{x^{m+1}}{(m+1!)}<\varepsilon\]

\oddelt
\begin{gather}
e_n=\sum_{k=0}^n \binom{n}{k} \left(\frac{x}{n}\right)^k=\sum_{k=0}^n \frac{n(n-1)\dots(n-k+1)}{n^k}\cdot\frac{x^k}{k!}=\sum_{k=0}^n c_{k,n}\frac{x^k}{k!}\\
\label{defexp1}\forall m\in\mathbb{N}_0 \forall n\geq m: \sum_{k=0}^m c_{k,n}\frac{x^k}{k!} \leq \sum_{k=0}^m \frac{x^k}{k!}=s_m\notag\\
\frac{x^{m+1}}{(m+1)!}-c_{m+1,n}\frac{x^{m+1}}{(m+1!)}<\varepsilon\tag{*}
\end{gather}

\begin{align}
a^2+b^2&=c^2\\
x&=y
\end{align}

\end{document}