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\begin{document}
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\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 3}

\noindent
\textbf{Posted: 2/7; Due: Friday, 2/13/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: 7.16-17 and 7.26-33. \smallskip

\begin{prob}
  Suppose $f: [a,b] \to \nr$ is continuous and
\[
\int_a^b x^n f(x) dx = 0
\]
for all integers $n \ge 0$. These integrals are called the
\emph{moments of $f$}. The conclusion addresses the question of
uniqueness in a \emph{moment problem}.
\begin{enumerate}
\item Evaluate $\int_a^b P(x) f(x) dx$ for any polynomial $P(x)$.
\item Prove that $\int_a^b (f(x))^2 dx = 0$.
\item Show that $f(x) = 0$ for all $x \in [a,b]$.
\end{enumerate}
\end{prob}

\begin{prob}
  Which of the following sequences $\{f_n\}$ of functions converges
  uniformly on $[0,1]$?
\begin{enumerate}
\item $f_n(x) = nx^2 (1-x)^n$;
\item $f_n(x) = n^2x (1-x^2)^n$;
\item $f_n(x) = n^2x^3 e^{-nx^2}$;
\item $\displaystyle{
f_n(x) = \frac{x^2}{x^2 + (1-nx)^2}.
}$
\end{enumerate}
\end{prob}

\begin{prob}
  Let $A = \{f \in \C([0,1], \nr) \;|\; \abs{f(x)} \le 1 \text{ for all
    $x \in [0,1]$}\}$.
\begin{enumerate}
\item Show that $A$ is a closed, bounded subset in $\C([0,1], \nr)$.
\item Show that the sequence $f_n(x) = x^n$ in $A$ does not have a
  convergent subsequence.
\item Explain why the fact that $A$ is closed and bounded (Part (1))
  does not contradict the fact that it is not compact, which follows
  from Part (2).
\end{enumerate}
\end{prob}

\begin{prob}
  Let $A$ be the same as in the preceding problem. Let $U_n := \{f \in
  \C([0,1], \nr) \;|\; \abs{f(0)- f(1/n)} < 1\}$ for $n \in \nn$. Show
  that $\{U_n\}$ is an open cover of $A$ but does not admit a finite
  subcover. (This is another blow to the Heine-Borel principle!)
\end{prob}

\begin{prob}
  A metric space is called \emph{separable} if it has a countable
  dense subset. For instance, $\nr$ is separable, because $\nq$ is
  countable and dense.
\begin{enumerate}
\item Prove that the metric space $\nc$ of complex numbers is
  separable.
\item Prove that $\C([0,1], \nr)$ is separable.
\end{enumerate}
\end{prob}

\begin{prob}
  Consider the set of polynomials $P(x)$ that have only terms of even
  degree, such as $x^6 - 3x^2 + 7$, but not $x+2$ or $2x^4-x^3 +4$.
\begin{enumerate}
\item Prove that these polynomials are dense in $\C([0,1], \nr)$.
\item Is this true for $\C([-1,1], \nr)$?
\end{enumerate}
\end{prob}

\end{document}