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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 10}}
\rhead{\large \textbf{DUE 11/19/2023}}
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{\Large\underline{\textbf{Exercises:}}}

\S $20, 21$

\begin{enumerate}

\item Let $X$ be a metric space with metric $d$. Prove the \textbf{reverse triangle inequality}: for all $x,y,z\in X$
	\[
		| d(x,y) - d(y,z) |\leq d(x,z).
	\]
	
	
\item Recall that the uniform metric on $\mathbb{R}^\mathbb{N}$ is defined as
	\[
		\overline{\rho}(\mathbf{x},\mathbf{y}) = \sup_{n\in \mathbb{N}} \overline{d}(x_n,y_n),
	\]	
where $\overline{d}(x_n,y_n)=\min\{ |x_n-y_n|,1\}$ is the standard bounded metric on $\mathbb{R}$.
	\begin{enumerate}
	\item Show that $\overline{\rho}$ is a metric.
	
	\item Let $C\subset \mathbb{R}^\mathbb{N}$ be the subset from Exercise 4 on Homework 8. Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the topology induced by $\overline{\rho}$.
	
	\item Let $h\colon \mathbb{R}^\mathbb{N}\to\mathbb{R}^\mathbb{N}$ be the function from Exercise 1 on Homework 9. Find necessary and sufficient conditions on the sequences $(a_n)_{n\in \mathbb{N}}, (b_n)_{n\in \mathbb{N}}$ which guarantee $h$ is continuous when $\mathbb{R}^\mathbb{N}$ has the topology induced by $\overline{\rho}$.
	
	\item For $\mathbf{x}\in \mathbb{R}^\mathbb{N}$ and $\epsilon>0$, show that
		\[
			U:=(x_1-\epsilon, x_1+\epsilon)\times (x_2-\epsilon, x_2+\epsilon) \times \cdots
		\]
	is \textbf{not} open with respect to the topology induced by $\overline{\rho}$.
	\end{enumerate}


\item Let $X$ be a metric space with metric $d$. For fixed $x_0\in X$, show that the function $f\colon X\to \mathbb{R}$ defined by $f(x)=d(x,x_0)$ is continuous.


\item For each $n\in \mathbb{N}$, define $f_n\colon\mathbb{R}\to\mathbb{R}$ by
	\[
		f_n(x) = \frac{1}{1+(x-n)^2}.
	\]
Show that the sequence of functions $(f_n)_{n\in \mathbb{N}}$ converges to the zero function pointwise but \textbf{not} uniformly.

\item Let $X$ be a set and let $Y$ be a metric space with metric $d$. Define a metric on $Y^X$ by
	\[
		\overline{\rho}( (y_x)_{x\in X}, (z_x)_{x\in X}) := \sup_{x\in X} \overline{d}(y_x,z_x),
	\]
where $\overline{d}(y,z) = \min\{ d(y,z),1\}$ is the standard bounded metric corresponding to $d$. Let $f_n,f\colon X\to Y$ be functions, $n\in \mathbb{N}$, and define $\mathbf{f_n}, \mathbf{f}\in Y^X$ by $\mathbf{f_n}:=(f_n(x))_{x\in X}$ and $\mathbf{f}:=(f(x))_{x\in X}$.
Show that $(f_n)_{n\in \mathbb{N}}$ converges uniformly to $f$ if and only if the sequence $(  \mathbf{f_n} )_{n\in \mathbb{N}}$ converges to $\mathbf{f}$ when $Y^X$ is given the topology induced by the metric $\overline{\rho}$.


\item[6*.] Let $\ell^2\subset \mathbb{R}^\mathbb{N}$ be the set of sequences $(x_n)_{n\in \mathbb{N}}$ for which the series $\sum_{n=1}^\infty x_n^2$ converges. For $\mathbf{x}=(x_n)_{n\in\mathbb{N}}\in \ell^2$ denote
		\[
			\|\mathbf{x}\|_2:= \left( \sum_{n=1}^\infty x_n^2 \right)^{1/2}.
		\]
	\begin{enumerate}
	\item For $\mathbf{x}\in \ell^2$ and $c\in \mathbb{R}$, show that $c\mathbf{x}\in \ell^2$ with $\|c\mathbf{x}\|_2 = |c| \|\mathbf{x}\|_2$.
	
	\item For $\mathbf{x},\mathbf{y}\in \ell^2$, show that the series $\sum_{n=1}^\infty |x_ny_n|$ converges and is bounded by $\|\mathbf{x}\|_2\|\mathbf{y}\|_2$.
	
	\item For $\mathbf{x},\mathbf{y}\in \ell^2$, show that $\mathbf{x}+\mathbf{y}\in \ell^2$ with $\|\mathbf{x}+\mathbf{y}\|_2\leq \|\mathbf{x}\|_2 + \|\mathbf{y}\|_2$.
	
	\item Show that $d_2(\mathbf{x},\mathbf{y})=\|\mathbf{x} - \mathbf{y}\|_2$ defines a metric on $\ell^2$.
	
	\item Show that the topology induced by $d_2$ is finer than the uniform topology but coarser than the box topology on $\ell^2$.
	\end{enumerate}
$^{*}$ Challenge Problem!

\end{enumerate}


	
\end{document}​