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\title{MTH435 - Homework 7}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 11/8/17. In this homework, the metrics $d_1$, $d_2$ and $d_\infty$ on $\R^2$ are defined as follows.
   \begin{align*}
   d_1((x_1,y_1),(x_2,y_2)) &= |x_1 - x_2| + |y_1 - y_2| \\
   d_2((x_1,y_1),(x_2,y_2)) &= \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2} \\
   d_\infty((x_1,y_1),(x_2,y_2)) &= \max \{ |x_1 - x_2|, |y_1 - y_2| \}
  \end{align*}
 On any non-empty set $X$, the discrete metric $\delta$ is defined as
 \[
  \delta(x,y) = \begin{cases}
                 0& \quad \text{if $x=y$}, \\
                 1& \quad \text{if $x \neq y$}.
                \end{cases}
 \]

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 \section{Warm-up Questions}
 
 \begin{que}
  Give examples of the following.
  \begin{itemize}
   \item[(i)] A decreasing sequence of positive reals $(x_n)$ such that $\sum (-1)^{n+1} x_n$ diverges.
   \item[(ii)] A sequence of positive reals $(x_n)$ such that $x_n \to 0$ but $\sum (-1)^{n+1} x_n$ diverges. 
  \end{itemize}

 \end{que}

 
 \begin{que}
  Draw $V_1((0,0))$ in $\R^2$ for the metrics $d_1$, $d_2$ and $d_\infty$.
 \end{que}  
  
\begin{que}
  Let $X$ be a metric space. Show that if $x \neq y$ in $X$ then there exist neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U \cap V = \varnothing.$\footnote{This is called the \emph{Hausdorff} property. You can remember the definition by noting that it says $x$ and $y$ can be ``housed off'' from each other.} 
 \end{que}
  
  \begin{que}
  Let $(X,d)$ be an arbitrary metric space. Decide if the following are metric spaces. 
   \begin{itemize}
    \item[(i)] $(X,d')$ where $d'(x,y) = \min \{d(x,y),1\}$.
    \item[(ii)] $(X,d')$ where $d'(x,y) = d(x,y)^2$.
    \item[(iii)] $(X,d')$ where $d'(x,y) = \frac{d(x,y)}{1+d(x,y)}$.
   \end{itemize}

  \end{que}

  
  
  \begin{que}
  Show that if $a=(x_1,y_1),b=(x_2,y_2) \in \R^2$ then
  \[
   \frac{1}{2}d_1(a,b) \leq \frac{1}{\sqrt{2}}d_2(a,b) \leq d_\infty(a,b) \leq d_2(a,b) \leq d_1(a,b). 
  \]
 \end{que}
 
 \begin{que}
  For the following pairs of subsets of $\R$, compute the values of  $\sup_{u \in U} \inf_{v \in V} |u-v|$ and $\sup_{v \in V} \inf_{u \in U} |u-v|$.
  \begin{itemize}
   \item[(i)] $U = \{0,1\}$, $V = [0,1]$.
   \item[(ii)] $U = \{0\}$, $V = \left\{ \frac{1}{n} \middle| n \in \N \right\} \cup \{0\}$.
   \item[(iii)] $U = [0,1]$, $V=[0,2]$.
  \end{itemize}
  
 \end{que}

 

 

 
 
 
 
 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Submitted Questions}
 
\begin{que}
Let $X$ be the set of sequences $(x_n)$ for which the associated series $\sum x_n$ is absolutely convergent and define $d \colon X \times X \to \R$ by $d((x_n),(y_n)) = \sum |x_n - y_n|$. Prove that $(X,d)$ is a metric space.
 \end{que}

 
 \begin{que}
  Let $((x_n,y_n))$ be a sequence in $\R^2$ and $(x,y) \in \R^2$. Prove the following are equivalent.
   \begin{enumerate}
    \item[(i)] $((x_n,y_n)) \to (x,y)$ in $(\R^2,d_1)$.
    \item[(ii)] $((x_n,y_n)) \to (x,y)$ in $(\R^2,d_2)$.
    \item[(iii)] $((x_n,y_n)) \to (x,y)$ in $(\R^2,d_\infty)$.
   \end{enumerate}
\end{que}

 \begin{que}\mbox{}
\begin{itemize}
 \item[(a)] Prove that if $((x_n,y_n)) \to (x,y)$ in $\R^2$ with the discrete metric then $((x_n,y_n)) \to (x,y)$ in $(\R^2,d_2)$.
 \item[(b)] Give an example of a sequence $((x_n,y_n))$ and a point $(x,y)$ such that $((x_n,y_n)) \to (x,y)$ in $(\R^2,d_2)$ but $((x_n,y_n))$ does not converge in $\R^2$ with the discrete metric. 
\end{itemize}
  
 \end{que}

 
 
 
 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
 Prove that the space of sequences $(x_n)$ such that $\sum x_n^2$ converges equipped with the metric defined by $d(x,y) = \sum (x_n - y_n)^2$ is a metric space.
 \end{que}
 
 
 \begin{que}
  Let $X = \{ V \subseteq \R \mid V \neq \varnothing, V \text{ is closed and bounded} \}.$ Prove that $d \colon X \times X \to \R$ defined by
  \[
   d(U,V) = \max \{ \sup_{u \in U} \inf_{v \in V} |u-v|, \sup_{v \in V} \inf_{u \in U} |u-v| \} 
  \]
is a metric on $X$.\footnote{This is called the \emph{Hausdorff} metric.}
 \end{que}
 
 \begin{que}
  Let $X$ be a space with a semi-metric $d$ (that is, $d$ satisfies all the properties of a metric except it is not necessarily true that $d(x,y)=0$ implies $x=y$.)
  \begin{itemize}
   \item[(i)] Prove that the relation $\sim$ on $X$ defined by $x \sim y$ if and only if $d(x,y) = 0$ is an equivalence relation on $X$.
   \item[(ii)] Let $X/ {\sim}$ be the set of equivalence classes of $\sim$. Prove that $(X/{\sim},d')$ is a metric space, where $d'([x],[y]) = d(x,y)$ (here, $[x]$ denotes the equivalence class of $x$ under $\sim$ - you will have to check $d'$ is well-defined).
  \end{itemize}

 \end{que}

 
 
 
 
 
 
 
 
 
 
 
 
 
 \end{document}