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\theoremstyle{definition}
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\newtheorem{defn}{Definition}
\newtheorem{notation}{Notation}
\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\PositiveRealNumbers}{{\mathbb{R}_+}}
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\begin{document}

\mytitle{Bounded uniformly continuous functions}

\medskip

\begin{objectives}
To study the basic properties of the $C^\ast$-algebra
of the bounded uniformly continuous functions on some metric space.
\end{objectives}

\begin{requirements}
Basic concepts of analysis: supremum, limit, continuity.
\end{requirements}

\bigskip

\begin{exercise}[definition of metric]
Let $X$ be a set and $\rho\colon X\times X\to\RealNumbers$ be a function.
The function $\rho$ is called \emph{metric} (or \emph{distance})
if the following conditions hold:
\begin{enumerate}
\item 
\item
\item For all $x\in X$,\quad $\rho(x,x)=0$.
\item For all $x,y\in X$ such that $x\ne y$,\quad $\rho(x,y)>0$.
\end{enumerate}
\end{exercise}

\bigskip

\section*{Modulus of continuity of a function}

\medskip
Denote by $\PositiveRealNumbers$ the set $(0,+\infty)$.

\medskip

\begin{defn}[modulus of continuity of a function]
Let $f\colon X\to\ComplexNumbers$.
Define the function $\omega_{\rho,f}\colon\PositiveRealNumbers\to[0,+\infty]$ by the following rule:
\[
\omega_{\rho,f}(\delta)\eqdef\sup\bigl\{|f(x)-f(y)|\colon\quad\rho(x,y)\le\delta\bigr\}.
\]
\end{defn}

\bigskip

\begin{exercise}[example]
Consider the set $[0,+\infty)$ with the usual metric $d(x,y)\eqdef|x-y|$.
Define $f\colon[0,+\infty)\to\ComplexNumbers$ by
\[
f(x)\eqdef\sqrt{x}.
\]
For all $\delta>0$ compute $\omega_{d,f}(\delta)$.
\end{exercise}

\clearpage
\section*{Monotonicity of the modulus of continuity}

\bigskip

\begin{notation}[subset of a set]
Let $A,B$ be some sets.
We write $A\subset B$ iff $A$ is a subset of $B$, that is,
iff every element of $A$ belongs to $B$.
Note that for every set $A$ the conclusion $A\subset A$ holds.
\end{notation}

\medskip

\begin{defn}[types of monotone functions] \mbox{}
Let $h\colon D\to[-\infty,+\infty]$ where $D\subset[-\infty,+\infty]$.
The function $h$ is called:
\begin{itemize}
\item \emph{increasing} iff\qquad
$\displaystyle\forall x,y\in D\qquad\bigl(x<y\qquad\Longrightarrow\qquad f(x)\le f(y)\bigr)$;
\item \emph{decreasing} iff\qquad
$\displaystyle\forall x,y\in D\qquad\bigl(x<y\qquad\Longrightarrow\qquad f(x)\ge f(y)\bigr)$;
\item \emph{strictly increasing} iff\qquad
$\displaystyle\forall x,y\in D\qquad\bigl(x<y\qquad\Longrightarrow\qquad f(x)<f(y)\bigr)$;
\item \emph{strictly decreasing} iff\qquad
$\displaystyle\forall x,y\in D\qquad\bigl(x<y\qquad\Longrightarrow\qquad f(x)>f(y)\bigr)$.
\end{itemize}
\end{defn}

\bigskip

\begin{exercise}
Let $f\colon X\to\ComplexNumbers$, $\delta_1,\delta_2>0$, $\delta_1<\delta_2$.
\begin{enumerate}
\item Compare the following subsets of $X\times X$:
\[
\bigl\{(x,y)\in X^2\colon\quad\rho(x,y)\le\delta_1\bigr\}
\qquad\underbrace{\hsp{3em}}_{\subset\ \text{or}\ \supset}\qquad
\bigl\{(x,y)\in X^2\colon\quad\rho(x,y)\le\delta_2\bigr\}.
\]
\item Compare the following subsets of $[0,+\infty]$:
\[
\bigl\{|f(x)-f(y)|\colon\quad\rho(x,y)\le\delta_1\bigr\}
\quad\underbrace{\hsp{3em}}_{\subset\ \text{or}\ \supset}\quad
\bigl\{|f(x)-f(y)|\colon\quad\rho(x,y)\le\delta_2\bigr\}.
\]
\item Compare the following numbers:
\[
\omega_{\rho,f}(\delta_1)
\qquad\underbrace{\hsp{3em}}_{\le\ \text{or}\ \ge}\qquad
\omega_{\rho,f}(\delta_2).
\]
\end{enumerate}
\end{exercise}

\bigskip

\begin{exercise}
Let $f\colon X\to\ComplexNumbers$.
From to the previous exercise make a conclusion
about the monotonicity of $\omega_{\rho,f}$.
\end{exercise}

\bigskip\bigskip\bigskip

\clearpage
\section*{Supremum-norm and bounded functions}

\bigskip

\begin{defn}[supremum-norm]
Given a function $f\colon X\to\ComplexNumbers$
denote by $\|f\|_\infty$ its supremum-norm:
\[
\|f\|_\infty\eqdef\sup_{x\in X}|f(x)|.
\]
\end{defn}

\bigskip

\begin{exercise}
Let $f,g\colon X\to\ComplexNumbers$ be bounded functions and $\lambda\in\ComplexNumbers$.
Recall exact formulas or upper bounds for the supremum-norm of the functions
$f+g$, $\lambda f$, $\overline{f}$ and $fg$
(just write formulas without any proofs):
\par\smallskip
\begin{align*}
\|f+g\|_\infty&\le\hsp{5em}
&
\|\lambda f\|_\infty&=\hsp{5em}
\\[3ex]
\|\overline{f}\|_\infty&
&
\|fg\|_\infty&
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Prove that the space of the bounded functions $X\to\ComplexNumbers$
is complete (with respect to the distance induced by the supremum-norm).
\end{exercise}


\clearpage
\section*{Arithmetic properties of the modulus of continuity}

\medskip

\begin{exercise}
Let $f,g\colon X\to\ComplexNumbers$ be bounded functions.
Find an upper bound of $\omega_{\rho,f+g}$
in terms of $\omega_{\rho,f}$ and $\omega_{\rho,g}$.
\end{exercise}

\vspace{6ex}

\begin{exercise}
Let $f\colon X\to\ComplexNumbers$ and $\lambda\in\ComplexNumbers$.
Express $\omega_{\rho,\lambda f}$ through $\omega_{\rho,f}$ and $\lambda$.
\end{exercise}

\vspace{6ex}

\begin{exercise}
Let $f\colon X\to\ComplexNumbers$.
Express $\omega_{\rho,\overline{f}}$ through $\omega_{\rho,f}$.
\end{exercise}

\vspace{6ex}

\begin{exercise}
Let $f,g\colon X\to\ComplexNumbers$ be bounded functions.
Find an upper bound of $\omega_{\rho,fg}$
in terms of $\omega_{\rho,f}$, $\omega_{\rho,g}$, $\|f\|_\infty$ and $\|g\|_\infty$.
\end{exercise}

\vspace{6ex}

\section*{Uniform approximation and modulus of continuity}

\bigskip

\begin{exercise}
Let $f,g\colon X\to\ComplexNumbers$ and $x,y\in X$.
Find an upper bound of $|f(x)-f(y)|$
in terms of $|g(x)-g(y)|$ and $\|f-g\|_\infty$.
\par\medskip
\begin{align*}
|f(x)-f(y)|
&=\bigl|f(x)-\ubrace{3em}+\ubrace{3em}-\ubrace{3em}+\ubrace{3em}-f(y)\bigr|
\\[2ex]
&\le
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Let $f,g\colon X\to\ComplexNumbers$.
Find an upper bound of $\omega_{\rho,f}$
in terms of $\omega_{\rho,g}$ and $\|f-g\|_\infty$:
\par\medskip
\[
\omega_{\rho,f}(\delta)\le \hsp{10em}
\]
\end{exercise}

\bigskip

\section*{Uniformly continuous functions}

\bigskip

\begin{defn}[uniformly continuous function]
A function $f\colon X\to\ComplexNumbers$ is called \emph{uniformly continuous} if
\[
\lim_{\delta\to0}\omega_{\rho,f}(\delta)=0.
\]
\end{defn}

\bigskip

\begin{exercise}
Write the previous definition using quantifiers $\forall$ and $\exists$:
\par\medskip
\[
\lim_{\delta\to0}\omega_{\rho,f}(\delta)=0
\qquad\xLeftrightarrow{\mbox{}\quad\text{def}\quad\mbox{}}\qquad
\forall\eps>0\qquad\hsp{17em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Simplify the previous definition using the fact that $\omega_{\rho,f}$ is monotone:
\[
\lim_{\delta\to0}\omega_{\rho,f}(\delta)=0
\qquad\Longleftrightarrow\qquad
\forall\eps>0\qquad\hsp{15em}
\]

\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\section*{Lipschitz-continuous functions}

\bigskip

\begin{exercise}
Write the definition of Lipschitz-continuous function in terms of $\omega_{\rho,f}$.
\end{exercise}

\vspace{12ex}

\begin{exercise}
Prove that every Lipschitz-continuous function is uniformly continuous.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\bigskip

\section*{Bounded uniformly continuous functions}

\bigskip

\begin{defn}
Denote by $\BUC(X,\rho,\ComplexNumbers)$
the set of the bounded uniformly continuous functions,
with usual arithmetic operations,
point-wise conjugation and supremum-norm.
Breve notation: $\BUC(X)$.
\end{defn}

\bigskip

\begin{exercise}
Prove that every uniformly continuous function $f\colon X\to\ComplexNumbers$ is continuous.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Prove that $\BUC(X)$ is closed under arithmetic operations and point-wise conjugation.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Prove that $\BUC(X)$ is a closed subspace of the space
of the bounded functions $X\to\ComplexNumbers$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Recall the definition of $C^\ast$-algebra.
Determine whether $\BUC(X)$ is a $C^\ast$-algebra or not.
\end{exercise}

\vspace{10ex}

\label{endlabel}
\end{document}
