%
%  This file was generated by vesti 0.2.8
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\documentclass[geometry,stix]{coprime}

\settitle{Real and Complex Analysis Rudin Solutions}{Sungbae Jeong}
\setgeometry{a4paper, margin = 1in}



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\def\supp{\mathop{\rm{supp}}}

\def\uc{{\frak c}}

\def\M{\Mf}

\def\Span{\mathop{\rm{span}}}

\def\dashfill{\leaders\hbox to 1em{\hss-\hss}\hfill}

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\def\Re{\mathop{\rm{Re}}}
\def\Im{\mathop{\rm{Im}}}
\def\sgn{\mathop{\rm{sgn}}}

\def\sing{\mathrel{\bot}}

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\let\liminf=\varliminf
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\let\limi=\liminf

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\def\dleq{\mathrel{\vcenter{\hbox{$\buildrel{\textstyle <}\over\sim$}}}}
\def\dgeq{\mathrel{\vcenter{\hbox{$\buildrel{\textstyle >}\over\sim$}}}}

\def\lp|#1|#2{%
    \def\lp@inner{#1}
%
    \def\lp@below{#2}
%
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\begin{document}
\section{Definitions}
\subsection{Specific Sets}

We denote several sets as follows:
\begin{table}[ht]
\centering
\begin{tabular}{*{2}{@{$\bullet$\kern5pt}r@{\kern5pt:\kern5pt}l@{\kern1cm}}}
    $\N$ & Set of natural numbers
    & $\Z_+$ & Set of positive integers \\
    $\Z$ & Sets of integers & $\Q$ & Sets of rational numbers \\
    $\R$ & Sets of real numbers & $\C$ & Sets of comples numbers \\
\end{tabular}
\end{table}

\subsection{Notations}
We define a notation $a\vee b\defeq\max\{a, b\}$,
$a\vee b\vee c = (a\vee b)\vee c$, and so on.
Similarly, $a\land b\defeq\min\{a, b\}$,
$a\land b\land c = (a\land b)\land c$, and so on.

If $X$ and $Y$ are sets, the set $Y^X$ be a collection of all
functions $f:X\to Y$.

If $\Phi$ be a collection of sets, then $\bigcup\Phi\defeq\bigcup_{E\in\Phi}E$
and similar for $\bigcap\Phi$.

The notation $A\cupdot B$ is an abbreviation of $A\cup B$ and
$A\cap B=\emptyset$.
Similarly, $\bigcupdot_1^\infty A_j$ is an abbreviation of
$\bigcup_1^\infty A_j$ and the collection $\{A_j\}_1^\infty$
is mutually disjoint.

In general, we denote a sequence by $(a_j)_1^\infty$ or $\{a_j\}_1^\infty$.
But in this paper, two notations are separated: The former is a notation of a sequence and the latter
denotes a {\it range} of a sequence $(a_j)_1^\infty$.
If the index of a given sequence is somewhat complicated, then use a script font to denote a sequence.

Let $\Fs$ be a sequence. Since sequence is a function from a subset of $\N$, the notation $\Fs_n$ is natural,
the $n^{\rm{th}}$ term.

In here, define $B(x,r)$ as an open ball centered at $x$ with a radius $r$ in a metric
space.\Footnote{In the textbook, it denotes an open ball as $B_r(x)$ but I will use $B(x,r)$.}
Denote $B[x,r]$ as a closed ball centered at $x$ with a radius $r$.

Let $X$ be a topological space and $E\subset X$. Then $\overline{E}$ denotes the \it{closure} of a set $E$,
$E^\circ$ denotes the \it{interior} of a set $E$ and $\partial E$ denotes the \it{boundary} of a set $E$.

If $\bf{P}(x)$ is a proposition with a variable $x$, write $\{x:\bf{P}(x)\}$ simply by $\{\bf{P}\}$.
If $\mu$ is a measure, we write $\mu\big(\{\bf{P}\}\big)$ simply by $\mu\{\bf{P}\}$.
For example, $\mu\{f>1\}$ means that $\mu\big(\{x:f(x)>1\}\big)$.

The notation $\converge f_n to f with {\rm something}$ describes the ``mode'' of convergence.
For example, If $f_n$ converges to $f$ in measure $\mu$, we denote $\converge f_n to f with \mu$.
If $f_n$ converges uniformly to $f$, then we denote $\converge f_n to f with \uniform$.
If $f_n$ converges to $f$ with respect to the metric $d$, we denote $\converge f_n to f with d$.

To refer some propositions and theorems in the main textbook, \it{Functional Analysis} by Rudin, I will
use the bold font like \bf{Theorem 1.1},
In this paper, some lemmas are presented. I will use the term \bf{Lemma L.1.1} 
to separate the lemmas in the textbook.
\def\convex#1{\mathop{\rm{co}}(#1)}


\end{document}
