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\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}
\def\Ra{\mathbb{Q}}
\def\Rl{\mathbb{R}}
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\begin{document}

\centerline{\textbf{Problem Set 1}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}


\bigskip\bigskip
\noindent
\textbf{Problem 1.} Let $(X,d)$ be a metric space.

\smallskip\noindent
(a) Prove the reverse triangle inequality: for every $x,y,z\in X$
\begin{align*}
d(x,y) &\ge \left\vert d(x,z) - d(z,y) \right\vert.
\end{align*}
(b) Prove that if $x_n \to x$ and $y_n \to y$ as $n\to \infty$,
then $d(x_n,y_n)\to d(x,y)$.



\bigskip
\noindent
\textbf{Problem 2.} Let $E$ be a finite set and let $P = \mathcal{P}(E)$ be the power set of $E$ (the set of all subsets of $E$). Define
$d : P \times P \to \Rl$ by
\[
d(A,B) = \text{card}(A\Delta B)
\]
where $\text{card}(A)$ is the number of elements of $A$ and
\[
A\Delta B = (A\setminus B) \cup (B\setminus A)
\]
is the symmetric difference of $A, B \subset E$. Show that $(P,d)$ is a metric space.

\bigskip
\noindent
\textbf{Problem 3.} If $(X,d)$ is a metric space, define $\rho : X\times X\to \Rl$ by
\[
\rho(x,y) = \frac{d(x,y)}{1 + d(x,y)}.
\]
(a) Show that $(X,\rho)$ is a metric space.

\smallskip\noindent
(b) Show that $(X,d)$ and $(X,\rho)$ have the same convergent sequences and the same metric topologies. Do they necessarily
have the same Cauchy sequences?

\smallskip\noindent
\textbf{Note.} Two metric spaces have the same metric topologies if they have the same open sets. A subset $G\subset X$ of a metric space $(X,d)$
is open if for every $x\in G$ there exists $\epsilon >0$ such that $B_\epsilon(x) \subset G$, where
$B_\epsilon(x) = \left\{y\in X : d(x,y) < \epsilon\right\}$.


\newpage
\noindent
\textbf{Problem 4.} Define $d : \Rl^2 \times \Rl^2 \to \Rl$ by
\[
d(x,y) = \sqrt{|x_1-y_1|} + \sqrt{|x_2-y_2|}\qquad x=(x_1,x_2),\quad y = (y_1,y_2).
\]
(a) Show that $(\Rl^2,d)$ is a metric space. Is this metric derived from a norm $\|\cdot\|$ on $\Rl^2$, meaning that
$d(x,y) = \|x-y\|$?

\smallskip\noindent
(b) Sketch the unit ball $B_1(0)$ in $(\Rl^2,d)$. Is it a convex set?


\bigskip
\noindent
\textbf{Problem 5.} Define the closure $\bar{A}$ of a subset $A\subset X$ of a metric space $X$ by
\[
\bar{A} = \bigcap\left\{ F \subset X : \text{$F\supset A$ and $F$ is closed}\right\}.
\]
Show that
\[
\bar{A} = \left\{ x\in X : \text{there exists a sequence $(x_n)$ with $x_n\in A$ and $x_n \to x$}\right\}.
\]
\textbf{Note.} A set $F\subset X$ is closed if $F^c = X\setminus F$ is open.

\bigskip
\noindent
\textbf{Problem 6.} Is the closure of the open ball
\[
B_r(x) = \left\{ y\in X : d(x,y) < r\right\}
\]
in a metric space $(X,d)$ always equal to the closed ball
\[
\bar{B}_r(x) = \left\{ y\in X : d(x,y) \le r\right\}?
\]



\bigskip
\noindent
\textbf{Problem 7.}
Let $X$ be the space of all real sequences of the form
\[
x = \left(x_1, x_2, x_3,\dots,x_N, 0, 0, \dots\right)\quad\text{for some $N\in \Nl$, where $x_n \in \Rl$},
\]
whose terms are zero from some point on. Define
\[
\|x\| = \max_{n\in \Nl} |x_n|.
\]
(a)  Show that $(X, \|\cdot\|)$ is a normed linear space (with
vector addition and scalar multiplication defined componentwise).

\smallskip\noindent
(b)
Show that $X$ is not complete.

\smallskip\noindent
(c) Let $c_0$ denote the space of all real sequences $(x_n)$ such that
$x_n\to 0$ as $n\to \infty$. Show that $(c_0, \|\cdot\|)$ is complete and $X$ is dense
in $c_0$.


\end{document}
