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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 9}}
\rhead{\large \textbf{DUE 11/12/2023}}
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{\Large\underline{\textbf{Exercises:}}}

\S $19, 20$

\begin{enumerate}

\item Let $(a_n)_{n\in \mathbb{N}},(b_n)_{\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ with $a_n>0$ for all $n\in \mathbb{N}$. Define a map $h\colon \mathbb{R}^\mathbb{N}\to \mathbb{R}^\mathbb{N}$ by
	\[
		h( (x_n)_{n\in \mathbb{N}}) = ( a_n x_n + b_n )_{n\in\mathbb{N}}.
	\]
	\begin{enumerate}
	\item Show that $h$ is a bijection.
	
	\item Show that if $\mathbb{R}^\mathbb{N}$ is given the product topology, then $h$ is a homeomorphism.
	
	\item Prove whether or not $h$ is a homeomorphism when $\mathbb{R}^\mathbb{N}$ is given the box topology.
	\end{enumerate}


\item For $\mathbf{x}=(x_1,\ldots, x_n), \mathbf{y}=(y_1,\ldots, y_n)\in\mathbb{R}^n$, define
	\[
		d_1(\mathbf{x},\mathbf{y}):= \sum_{j=1}^n |x_j - y_j|.
	\]
	\begin{enumerate}
	\item Show that $d_1$ is a metric on $\mathbb{R}^n$.
	
	\item Show that the topology induced by $d_1$ equals the product topology on $\mathbb{R}^n$.
	
	\item For $n=2$ and $\mathbf{0}=(0,0)\in \mathbb{R}^2$, draw a picture of $B_{d_1}(\mathbf{0},1)$.
	\end{enumerate}


	
\item Let $X$ be a metric space with metric $d$. For $x\in X$ and $\epsilon>0$, show that $\{y\in X\mid d(x,y)\leq \epsilon\}$ is a closed set.

\item Let $X$ be a metric space with metric $d$. Show that $d\colon X\times X\to \mathbb{R}$ is continuous.	


\item For $\mathbf{x}=(x_1,\ldots, x_n), \mathbf{y}=(y_1,\ldots, y_n)\in\mathbb{R}^n$ and $c\in \mathbb{R}$ define
	\begin{align*}
		\mathbf{x}+\mathbf{y}&:=(x_1+y_1,\ldots, x_n+y_n),\\
		c\mathbf{x} &:= (c x_1,\ldots, cx_n),\\
		\mathbf{x}\cdot\mathbf{y}&:=x_1y_1+\cdots +x_ny_n,\\
		\|\mathbf{x}\|&:=(x_1^2+\cdots +x_n^2)^{1/2}.
	\end{align*}
	\begin{enumerate}
	\item For $\mathbf{x},\mathbf{y},\mathbf{z}\in \mathbb{R}^n$ and $a,b\in \mathbb{R}$, prove the following formulas
		\begin{align*}
			\|\mathbf{x}\|^2 &= \mathbf{x}\cdot\mathbf{x}\\
			(a\mathbf{x})\cdot (b\mathbf{y})&=(ab)(\mathbf{x}\cdot\mathbf{y})\\
			\mathbf{x}\cdot\mathbf{y}&=\mathbf{y}\cdot\mathbf{x}\\
			\mathbf{x}\cdot(\mathbf{y}+\mathbf{z}) &= \mathbf{x}\cdot \mathbf{y}+\mathbf{x}\cdot \mathbf{z}
		\end{align*}
	
	\item Show that $|\mathbf{x}\cdot\mathbf{y}| \leq \|\mathbf{x}\| \|\mathbf{y}\|$. [\textbf{Hint:} for $\mathbf{x},\mathbf{y}\neq 0$ let $a=\frac{1}{\|\mathbf{x}\|}$ and $b=\frac{1}{\|\mathbf{y}\|}$ and use the fact that $\| a\mathbf{x} \pm b\mathbf{y}\|^2\geq 0$.]
	
	\item Show that $\|\mathbf{x}+\mathbf{y}\| \leq \|\mathbf{x}\|+\|\mathbf{y}\|$.
	
	\item Prove that the euclidean metric $d(\mathbf{x},\mathbf{y}):=\|\mathbf{x} - \mathbf{y}\|$ is indeed a metric.
	\end{enumerate}	



\item[6*.] For $\mathbf{x}=(x_1,\ldots, x_n)\in\mathbb{R}^n$ and $1\leq p <\infty$, define
	\[
		\| \mathbf{x}\|_p:=(|x_1|^p+\cdots +|x_n|^p)^{1/p},
	\]
and for $p=\infty$ define
	\[
		\|\mathbf{x}\|_\infty:= \max \{|x_1|, \ldots, |x_n|\}.
	\]
In this exercise you will show $d_p(\mathbf{x},\mathbf{y}):=\| \mathbf{x} - \mathbf{y}\|_p$ defines a metric for each $1\leq p\leq \infty$. Observe that $p=1,2,\infty$ yield the metric from Exercise 2, the euclidean metric, and the square metric, respectively.
	\begin{enumerate}
	\item For $1<p<\infty$, show that if $q>0$ satisfies $\frac{1}{p}+\frac{1}{q}=1$ then $1<q<\infty$. We call $q$ the \textbf{conjugate exponent} to $p$.
	
	\item For $a,b\geq 0$ and $0<\lambda <1$, show that $a^\lambda b^{1-\lambda} \leq \lambda a + (1-\lambda) b$.
	
	\item Prove \textbf{H\"older's Inequality}: for $1<p<\infty$ with conjugate exponent $q$ and $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$ show that
		\[
			|x_1y_1|+\cdots +|x_ny_n| \leq \|\mathbf{x}\|_p \|\mathbf{y}\|_q.
		\]
	
	\item Prove \textbf{Minkowski's Inequality}:  for $1<p<\infty$ and $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$ show that
		\[
			\|\mathbf{x}+\mathbf{y}\|_p \leq \|\mathbf{x}\|_p + \|\mathbf{y}\|_p.
		\]
	[\textbf{Hint:} use $|x_j+y_j|^p \leq (|x_j|+|y_j|)|x_j+y_j|^{p-1}$.]
	
	\item Show that $d_p$ is a metric for $1< p <\infty$.
	
	\item Show that the topology induced by $d_p$ equals the product topology on $\mathbb{R}^n$ for $1<p<\infty$, where $\mathbb{R}$ has the standard topology. [\textbf{Hint:} show that $\|\mathbf{x}\|_\infty\leq \|\mathbf{x}\|_p\leq \|\mathbf{x}\|_1$.]
	\end{enumerate}
$^{*}$ Challenge Problem!

\end{enumerate}


	
\end{document}​