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\begin{document}



\title{ Math 3220-3 Homework 1}
\author{Due January 23}
\maketitle

\begin{enumerate}  \item Let $f_n(x) = x^n\in\cC([0,1])$ 
    \begin{enumerate}
  \item Prove that the sequence $\{f_n\}$ has no convergernt subsequence (in the norm of $\cC([0,1])$). \vskip3pt
  \item Use this to prove that the unit ball $\{f\in\cC([0,1]) : ||f||\le 1\}$ is not compact. 
\end{enumerate}
\vskip10pt
  \item 
  \begin{enumerate}
  \item Let $(X,d)$ be a metric space and let $K\subset X$ be a compact subset.  Prove that for all $\epsilon >0$ there are finitely many points $x_1,\dots,x_n\in K$ so that, for every $x\in K$ there exists an $i, i= 1,\dots,n$, such that $d(x,x_i) < \epsilon$
  \vskip3pt
  \item Use this to prove that, if $K\subset \cC([0,1])$ is compact, then $K$ is equicontinuous.
  
\end{enumerate}
\vskip 10pt
  \item Recall the norms $||f||_1 = \int_0^1|f(x)| \ dx$ and $||f||_\infty= \sup_{x\in X} \{|f(x|)\}$ on $\cC([0,1])$.
  \vskip3pt
  \begin{enumerate}
  \item Prove that $||f||_1\le \||f||_\infty$ for all $f\in \cC([0,1])$
  \vskip 3pt
  \item Prove that there is no constant $C>0$ such that $||f||_\infty \le C ||f||_1$ holds for all $f\in \cC([0,1])$ by producing a sequence $f_n\in\cC([0,1])$ with $||f_n||_\infty \to \infty$ and $||f_n||_1 = 1$
  \item Prove that $\cC([0,1])$ with norm $||f||_1$ is not a complete metric space.  Observe that this gives another proof of (b).
  
  
  \emph{Suggestion} Consider (in $\cC([-1,1])$ for simpler formulas) the sequence 
  $$
  f_n(x) = \begin{cases}
     -1 & \text{ if }-1\le x\le 1/n, \\
    nx  & \text{if} -1/n \le x\le 1/n, \\
    1 & \text{ if } 1/n \le x\le 1.
    
\end{cases} 
$$ 
    \vskip10pt
 
\end{enumerate}
\item This problem and the next are part of Rudin,  Chapter 7, Exercise 4. Let
$$
f(x) = \sum_{n=1}^\infty \frac{1}{1+n^2 x}\
$$
\begin{enumerate}
\item
For which $x\in\R$ does the series converge?  

\emph{Note}: if for a given $x\in \R$ some term in the series is not defined, then the series does not converge for that $x$.
\item For which $x\in\R$ does it converge absolutely?  
\item Is $f$ bounded?
\end{enumerate}
\vskip10pt
\item Using the same series as in the last problem
\begin{enumerate} 
\item For which intervals in $\R$ does the series converge uniformly?  
\item For which intervals in $\R$ does the series converge, but not uniformly?.
\end{enumerate}
\end{enumerate}


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