\documentclass[12pt]{article}
\usepackage{amsfonts,amsmath}

\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}
\def\Ra{\mathbb{Q}}
\def\Rl{\mathbb{R}}
\def\Ts{\mathbb{T}}
\def\Ir{\mathbb{Z}}

\begin{document}

\centerline{\textbf{Problem Set 4}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}

\bigskip\bigskip
\noindent
\textbf{Problem 1.} Let $f : X \to Y$ be a one-to-one, onto map between metric spaces $X$, $Y$.


\smallskip\noindent
(a) If $f$ is continuous and $X$ is compact, prove that $f$ is a homeomorphism. Does this result remain true
if $X$ is not compact?

\smallskip\noindent
(b) Suppose that $f$ is uniformly continuous and $f^{-1}$ is continuous. If $Y$ is complete, prove that $X$ is complete. If $X$ is complete, does it follow that $Y$ is complete?

\bigskip
\noindent
\textbf{Problem 2.} Consider the space of continuously differentiable functions,
\[
C^{1}\left([0,1]\right) = \left\{ f : [0,1]\to\Rl \mid
\text{$f$, $f'$ are continuous}\right\},
\]
with the $C^{1}$-norm,
\[
\|f\|_{C^1} = \sup_{0\le x \le 1} |f(x)| + \sup_{0\le x \le 1} |f'(x)|.
\]
Prove that $C^{1}\left([0,1]\right)$ is a Banach space.



\bigskip\noindent
\textbf{Problem 3.} (a) Suppose that $f, g\in C([a,b])$. Prove the Cauchy-Schwarz inequality:
\[
\left|\int_a^b f(x) g(x) \, dx\right| \le \left(\int |f(x)|^2\, dx\right)^{1/2} \left(\int |g(x)|^2\, dx\right)^{1/2}.
\]
(b) Let
\[
\mathcal{F} = \left\{ f\in C^1([0,1]) : \text{$f(0) = f(1) =0$, $\int_0^1 f'(x)^2\, dx \le 1$}\right\}.
\]
Prove that $\mathcal{F}$ is precompact in $\left(C([0,1]), \|\cdot\|_\infty\right)$.




\newpage
\noindent
\textbf{Problem 4.} (a) If $f : \Rl\to \Rl$ is continuous, show that
\[
\frac{1}{h}\int_{x}^{x+h} f(t)\, dt \to f(x)\qquad \text{as $h\to 0$}.
\]
(b) Given $f\in C([0,1])$, extend $f$ to a function $\tilde{f} : \Rl \to \Rl$ by
\[
\tilde{f}(x) = \begin{cases} f(1) &\text{if $x>1$}
\\
f(x) &\text{if $0\le x \le 1$}
\\
f(0) &\text{if $x < 0$}
\end{cases}
\]
Define $f_h : [0,1] \to \Rl$ for $h\in \Rl$ by
\[
f_h(x) = \frac{1}{h}\int_{x}^{x+h} \tilde{f}(t)\, dt.
\]
Show that $f_h \in C^1([0,1])$. What is $f_h'$?

\smallskip\noindent
(c) Show that $f_h \to f$ uniformly as $h\to 0$, meaning that $C^1([0,1])$ is dense in $\left(C([0,1]), \|\cdot\|_\infty\right)$.%%



\bigskip
\noindent
\textbf{Problem 5.}  Let $X$ be a metric space and let $B(X)$ denote the normed linear space of all bounded (but not necessarily
continuous) functions $f : X \to \Rl$ with the sup-norm
\[
\|f\| = \sup_{x\in X} |f(x)|.
\]
Fix $x_0\in X$, and for
each $a\in X$ define $\phi_a : X \to \Rl$ by
\[
\phi_a(x) = d(x,a) - d(x,x_0).
\]
Finally, define $\Phi : X \to B(X)$ by $\Phi(a) = \phi_a$.


\smallskip\noindent
(a) Show that $B(X)$ is complete.

\smallskip\noindent
(b) Show that $\Phi$ is an isometric embedding of $X$ into $B(X)$.

\smallskip\noindent
(c) Use this result to show that every metric space $X$ has a completion.



\bigskip
\noindent
\textbf{Problem 6.} Suppose that $A\subset X$ is a dense subset of a metric space $X$ and $f : A \to Y$ is a uniformly
continuous function into a complete metric space $Y$. Prove that there is a unique continuous extension  $\tilde{f} : X \to Y$ of $f$,
such that $\left.\tilde{f}\right|_A = f$.


\end{document}
