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{{\noindent Math 5510} \hfill {Homework 1--  Due Wednesday 8/30/17
\hfill {\ \ \ \ }\hfill}



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\begin{enumerate}
\item Here is another proof of the Cauchy-Schwarz inequality (where $u=(u_1,\dots,u_n)$, $v=(v_1,\dots,v_n)\in R^n$, with the usual dot product $u\cdot v = u_1v_1+ \dots + u_nv_n$).
 \begin{equation}
\label{cs0}
|u\cdot v|\le |u||v| \ \ (\hbox{with equality } \Leftrightarrow \{u,v\} \hbox{ linearly dependent} )
\end{equation}
 
 Squaring both sides and taking their difference (\ref{cs0}) is equivalent to
\begin{equation}
\label{cs1}
|u|^2 |v|^2 - (u\cdot v)^2 \ge 0 \ \ (\hbox{equality } \Leftrightarrow \{u,v\} \hbox{ linearly dependent} )
\end{equation}


\begin{enumerate}
  \item For $n=2$ the derive the identity  
  \begin{equation}
  \label{cs2}
|u|^2|v|^2 - (u\cdot v)^2 = (u_1 v_2 - u_2 v_1)^2
\end{equation}
   and show that (\ref{cs2}) implies the inequality (\ref{cs1}) including the case of equality.
  
  \item Derive the identity similar to (\ref{cs2}) for arbitrary $n\ge 2$ and use it to derive the Cauchy Schwarz inequality, including the case of equality.  If you don't see what the identity is for arbitrary $n$, do it for $n = 3$.
 \end{enumerate}


\item Let $f:[0,\infty)\to [0,\infty)$ be a function that satisfies: 
\begin{enumerate}
\item $f(0) = 0$.
\item $f$ is strictly increasing: $f(s) < f(t) $ whenever $0\le s < t $.
\item $f(s+t)\le f(s) + f(t)$ for all $s,t\ge 0$.
\end{enumerate}

Prove that If $(X,d)$ is a metric space, $(X,f\circ d)$ is also a metric space.

\item
\begin{enumerate}
  \item Suppose $(X,d)$ is a metric space.  Prove that so is
  $$
  \Big{(}X,\frac{d}{1+d}\Big{)}.
  $$

  \item Which of the following are metrics on $\R$?
  
  (i) \  $d(x,y) = |x-y|^2$ \ \ \ \  (ii) \  $d(x,y) = \sqrt{|x-y|}$
 \end{enumerate}

\item Let 
$$
d_{(2)}(x,y)=\sqrt{(x_1 - y_1)^2 + \dots + (x_n - y_n)^2},\ \ d_{(1)}(x,y)= |x_1 - y_1| + \dots + |x_n - y_n|
$$ 
be the Euclidean and  \lq\lq taxicab" distance functions on $\R^n$, respectively,  Prove the following two inequalities that compare the two distances and discuss the case of equality in each inequality.
$$
d_{(2)}(x,y) \le d_{(1)}(x,y)\le \sqrt{n} \ d_{(2)}(x,y).
$$
For the case $n=2$ draw pictures that illustrate the inequalities and the case of equality.
 \end{enumerate}  
   
 


                

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