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\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{7}{Complex geometry handout 7: Stone-Weierstrass approximation theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weierstrass approximation theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M$ be a topological space, and
$\|f\|:= \sup_M |f|$ {\bf the sup-norm on functions}.
{$C^0$-topology} on the space $C^0(M)$ of continuous functions
is topology defined by the sup-norm.
\ed

\exercise
%Assume that $M$ is compact.
Prove that $C^0M$ with the metric defined by
the sup-norm is a complete metric space.
\ez

\exercise
(``Dini's theorem'')\\
Let $\{f_i\}$ be a sequence of bounded
continuous functions on a compact space $M$,
and suppose that $f_i(t)\geq f_{i-1}(t)$ for all $t$ and $i$.
Suppose that $\lim_i f_i(t)=f(t)$ for
some continuous function $f$. Prove that
the sequence $\{f_i(t)\}$ converges to $f(t)$ uniformly.
\ez

\exercise
\label{_module_approx_Zadacha_}
Consider the sequence $P_i$, $i=0, 1, 2, ...$
of polynomials on $[0,1]$
determined recursively as follows: $P_0(t)=0$, and
$P_i(t)= P_{i-1}(t) + \frac 1 2 (t-P_{i-1}(t)^2)$.
For all $t\in[0,1]$ and
  all $i=1, 2,..., $, prove the following.
\enum
\item
  Prove that $0\leq P_i(t) \leq \sqrt t$.
\item
  Prove that $P_i(t)\geq P_{i-1}(t)$.
\item
  Prove that $\{P_i(t)\}$ converges 
  pointwisely to $\sqrt t$ on $[0,1]$.
\item
  Prove that $\{P_i(t)\}$ converges 
  uniformly to $\sqrt t$ on $[0,1]$
\item
  Prove that $Q_i(t):= P_i(t^2)$
  converges uniformly to $|t|$ on $[-1,1]$.
\ee
\ez

\exercise
Let $F(t)$ be a piecewise linear, continuous
function on $[a,b]\subset \R$. Prove that
$F(t)$ can be expressed as a sum
$\sum_{i=0}^n \alpha_i |x-c_i|$ for
some $\alpha_i, c_i$.
\ez

\exercise
Prove that any piecewise linear, continuous
function on $[a,b]\subset \R$ can be obtained
as a uniform limit of polynomials.
\ez

\exercise {\bf (Weierstrass approximation theorem)}\\
Prove that any continuous function on
 $[a,b]\subset \R$ admits a uniform approximation
 by polynomials.
\ez

\remark This particular proof of Weierstrass approximation
is due to Lebesgue.
\er

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stone-Weierstrass approximation theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From now on we assume that $M$ is compact, Hausdorff topological space.


\definition
Let  $A\subset C^0 M$ be a subspace in the space of
continuous functions. We say that $A$ {\bf separates the points} of $M$
if for all distinct points $x, y\in M$, there exists $f\in A$ such that
$f(x) \neq f(y)$.
\ed

\exercise
Let $A\subset  C^0 M$ be a subring, and
$\bar A$ its closure in $C^0$-topology.
\enum \item Prove that for any $a\in A$, the function
$|a|$ belongs to $\bar A$.
\item Prove that for any $a, b\in A$,
the function $\min(a,b)$ belongs to $\bar A$.
\ee
\ez

\hint Use Exercise \ref{_module_approx_Zadacha_}.
\eh

\exercise
Let  $A\subset  C^0 M$ be a subring separating points,
$\bar A$ its closure, and $U\ni x$ a neighbourhood of $x\in M$.
Prove that for any $\epsilon >0$ there exists
$a \in  \bar A$ taking values in $[0,1]$ such that
$a(x)=1$ and $a\restrict{M\backslash U} < \epsilon$.
\ez

\hint
Find a finite covering of the compact $M\backslash U$
by open sets $U_i$ and functions $f_i \in \bar A$
such that $f_i(x)=1$ and $f_i\restrict U_i < \epsilon$,
and put $a:= \min_i(f_i)$.
\eh

\exercise
Let  $A\subset  C^0 M$ be a subring separating points,
$\bar A$ its closure, and $f\in C^0(M)$ any function. Prove that
for all $x\in M$ there exists a function
 $f_x\in \bar A$ such that $f_x\leq f$ and $f_x(x) > f(x)-\epsilon$.
\ez

\hint Use the previous exercise.
\eh

\exercise {\bf (Stone-Weierstrass theorem)}\\
Let $A\subset  C^0 M$ be a subring separating points,
and $\bar A$ its closure. Prove that  $\bar A= C^0M$.
\ez

\hint Use the previous exercise and find
a neighbourhood $U_x$ and a function $f_x\leq f$
such that $(f_x+\epsilon)\restrict{U_x}  >f\restrict{U_x}$.
Find a finite covering $\{U_{x_i}\}$ by such $U_x$, such that
$f \geq \max_i f_{x_i}> f-\epsilon$.
\eh

 
\end{document}
