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\centerline{\Large Math 675 Homework 2}
\centerline{Due 9/05/2018}
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Responses must be typed, with a two-page maximum and LaTeX highly recommended. Grade will be based on style (2 points) and all of the problems.

\begin{enumerate}[1.]
\item (2 points) Prove the Cauchy-Schwartz inequality, following Problem 2 on Page 45.
\item (2 points each) Read through the proof that, for $p\in [1,\infty)$, $d_p$ is a metric on $\R^n$ (Example 10 starting on Page 41). 
\begin{enumerate}[(a)]
\item Explain why the proof fails for $p\in (0,1)$. 
\item Use an example to show that $d_{1/2}$ is not a metric in $\R^2$.
\end{enumerate}

\item (2 points each) Show that in $\R^n$ the metric $d_\infty$ is the limit of the functions $d_p$ as $p\rightarrow \infty$:
\begin{enumerate}[(a)]
\item Experimentally by drawing a sequence of unit balls in $\R^2$. (Hint: use Wolfram Alpha or other software, and then include the graphics into your write-up.)
\item Provide a proof that $$\max |x_i-y_i|=\lim_{p\rightarrow \infty}\left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}.$$
\end{enumerate}
\item (2 points) Find an isometry (see Definition 2 on p.44) between $C_\infty[a,b]$ and $C_\infty[c,d]$ for any $a<b$ and $c<d$.
\end{enumerate}





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