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%version 1.0,\ \   06.02.2018 (SW part from the Ergodic 2017 assign-01)
%version 1.1,\ \   07.02.2018 melkie ispravleniya ot K. A. 
%version 1.2,\ \   20.02.2018 added a hint to H^*=H.


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


\begin{document}

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\listok{3}{Hodge theory 3: Stone-Weierstrass theorem and Banach spaces}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\scriptsize
  {\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra stuff to work.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 bounded 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 
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 every points $x\neq y\in M$, there exists $f\in A$ such that
$f(x) \neq f(y)$.
\ed

\exercise
Let $A\subset  C^0 M$ be an $\R$-subalgebra, 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 $f_i \in \bar A$ taking values in $[0,1]$
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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Banach spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
A non-negative real-valued function $v\arrow |v|$
on a vector space is called {\bf norm} if
\begin{itemize}
\item $|v|=0$ if and only if $v=0$
\item For each $r\in \R$, $|rv| = |r||v|$.
\item $|v_1+v_2| \leq |v_1| + |v_2|$.
\end{itemize}
Clearly, norm defines a metric on $V$,
with $d(x, y):= |x-y|$. A norm is {\bf complete}
if this metric is complete.
\ed

\definition
A complete normed topological vector 
space is called {\bf Banach space}.
\ed

\definition
Let $V_1$, $V_2$ be vector spaces equipped with a norm.
{\bf Norm} (``operator norm'')
of a linear operator $E:\; V_1 \arrow V_2$ 
is the number $\|E\|:=\sup_{v\neq 0} \frac {|E(v)|}{|v|}$.
An operator with finite norm is called {\bf bounded}.
\ed

\exercise[!]
Show that $E$ is continuous
if and only if it is bounded.
\ez

\exercise
Prove that $\|E\|$ defines a norm on the space
of bounded operators $E:\; V_1 \arrow V_2$. 
\ez


\exercise[!]
Suppose that $V_1, V_2$ are Banach spaces,
and $\Hom(V_1, V_2)$ the space of bounded operators
equipped with the operator norm.
Prove that $\Hom(V_1, V_2)$
is a Banach space.
\ez

\remark
From now on, all linear operators on topological vector spaces
are assumed continuous, unless stipulated otherwise.
\er

\definition
Let $H$ be an infinite-dimensional vector space
equipped with a positive-definite scalar product. 
We say that $H$ is a {\bf Hilbert space} if it is
 complete and contains a countable dense set.
\ed

\exercise[!]
Let $g$ denote the scalar product on a Hilbert space $H$.
Prove that $g$ defines an isomorphism from $H$ to $H^*$,
whete $H^*$ denotes the space of continuous functionals.
\ez

\hint
Let $x_i$ be the orthonormal basis in $H$.
Use the dual basis to write any form as $\lambda=\sum_i \lambda_i e_i$.
Prove that $\frac{|\lambda(x)|^2}{|x|^2}= \sum |\lambda_i|^2$.
\eh

\exercise[*]
Prove that a sphere in a Hilbert space is contractible.
\ez

\exercise[*]
Consider the group $GL(H)$ of 
linear automorphisms of a Hilbert space, 
taken with the norm topology.
Prove that it is contractible.
\ez

\exercise[**]
Consider the group $O(H)$ of 
linear isometries of a Hilbert space, 
taken with the norm topology.
Prove that it is contractible.
\ez

\exercise
\enum
\ite Let $H$ be a Hilbert space.
Prove that the closure of the unit ball is non-compact.
\ite[*]
Let $V$ be an infinite-dimensional Banach space.
Prove that the closure of the unit ball is non-compact.
\ee
\ez

\exercise[**]
Let $E;\; V_1\arrow V_2$ be a bijective, continuous linear operator
on Hilbert spaces. Prove that $E^{-1}$ is bounded.
\ez 

\definition
Let $V$ be a vector space, and $|\ |_1$, $|\ |_2$ - 
norms on $V$. We say that these norms are {\bf equivalent}
if the identity operator $(H, |\ |_1) \arrow (V, |\ |_2)$
is a homeomorphism. 
\ed

\exercise[!]
Prove that
$|\ |_1$, $|\ |_2$ are equivalent if and only if
there exists Á constant $C>1$ such that for all $v\in V$
one has $C^{-1} |v |_1 \leq |v |_2\leq C |v |_1$.
\ez

\exercise
Prove that on a finite-dimensional space all norms are 
equivalent.
\ez


\end{document}
