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\title{Homework 10 for 18.100B/C, Fall 2010\\
Due Thursday 2 December}
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%\author{Richard Melrose}
%\address{Department of Mathematics, Massachusetts Institute of Technology}
%\email{rbm@math.mit.edu}
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As usual, homework is due in 2-108 by 11AM on Thursday 2 December, or by
email before 5PM on the same day. This is the last homework for
this course!

\begin{enumerate}
\item Let $E$ be a set and let $Y$ be a metric space. Consider all the
  bounded maps $f:E\longrightarrow Y$ -- so for each such map there is a
  constant $M$ and a point $y\in Y$ with the property that $d_Y(f(e),y)\le M$
  for all $e\in E.$ Let $\mathcal{B}(E;Y)$ be the set of these bounded maps
  and define for each $f,$ $g\in\mathcal{B}(E;Y)$ 
$$
d(f,g)=\sup_{e\in E}d_Y(f(e),g(e)).
$$
Show that this is a metric on $\mathcal{B}(E;Y)$ and that if $Y$ is complete
so is $\mathcal{B}(E;Y).$
\item Show that if $X$ is a metric space and $Y$ is a (complete if you want
  because of the previous question but it is not needed here) metric space
  then the set $\mathcal{C}(X;Y)$ of continuous and bounded maps
  $f:X\longrightarrow Y$ is a closed subset of $\mathcal{B}(X;Y)$ defined
  in the previous question.
\item Rudin Chap 7 No 24. Let $X$ be a metric space with metric $d$ and
  suppose $a\in X$ is a fixed point. Assign to each $p\in X$ the function
  $f_p:X\longrightarrow \mathbb{R}$ where $f_p(x)=d(p,x)-d(a,x)$ for each
  $x\in X.$ Show that $f_p\in\mathcal{C}(X),$ i.e.\ it is a continuous
  bounded function, and so this construction defines a map $\Phi:X\longrightarrow
  \mathcal{C}(X).$ Prove that
$$
\sup_{x\in X}|f_p(x)-f_q(x)|=d(p,q)\ \forall\ p,q\in X.
$$
Hence conclude that $\Phi$ is an \emph{isometry} (a map between metric
spaces preserving the distance) which maps $X$ 1-1 onto a subset of
$\mathcal{C}(X).$ Since the latter is complete deduce that the closure of
the range of $\Phi$ is a metric completion of $X$ -- is a complete metric
space which has $X$ (represented as the range of $\Phi)$ as a dense subset and
which has a distance which restricts to the distance on $X.$
\item
Let $f_n:[0,1]\longrightarrow [0,\infty)$ be a sequence of continuous
  functions which is pointwise decreasing with $\lim_{n\to\infty}f_n(x)=0$
  for each $x\in[0,1].$ Show that $\lim_{n\to\infty}\int_0^1f_n(x)dx=0.$  
\item
Suppose that $f:[0,1]\longrightarrow \mathbb{R}$ is a Riemann integrable
function. Show that, as a consequence of the Stone-Weierstrass theorem,
there is a sequence of polynomials $p_n$ such that
$$
\lim_{n\to\infty}\int_0^1|f(x)-p_n(x)|dx=0.
$$
Hint: Use a problem from earlier homework about Riemann integrable functions.
\end{enumerate}

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