
\documentclass{slides}

\usepackage{amssymb, amsmath, amscd, color, epsfig}
%\usepackage[matrix,arrow]{xy}

\newcommand{\green}{\color[rgb]{0,0.4,0}}
\newcommand{\purple}{\color[rgb]{0.4,0,0.4}}
\newcommand{\red}{\color[rgb]{0.7,0,0}}
\newcommand{\blue}{\color{blue}}


\def\eqref#1{(\ref{#1})}
\newcommand{\goth}{\mathfrak}
\newcommand{\g}{{\frak g}}
\newcommand{\arrow}{{\:\longrightarrow\:}}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{#1}}\right.}}
\newcommand{\cntrct}                % contraction with a vector field
{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}


\def\Bbb#1{\mathbb #1}


\newcommand{\calo}{{\cal O}}
\newcommand{\cac}{{\cal C}}

% Correcting TeX...
%\let\oldtilde=\tilde
%\renewcommand{\tilde}{\widetilde}
\renewcommand{\bar}{\overline}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}

% Operatornames
\newcommand{\even}{{\rm even}}
\newcommand{\ev}{{\rm even}}
\newcommand{\odd}{{\rm odd}}
\newcommand{\const}{{\sf const}}
\newcommand{\fl}{{\rm fl}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Hol}{\operatorname{{\cal H}ol}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\id}{\operatorname{\text{\sf id}}}
\newcommand{\symb}{\operatorname{\text{\sf symb}}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Var}{\operatorname{Var}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Can}{\operatorname{Can}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}

\newcommand{\coker}{\operatorname{coker}}
\newcommand{\ind}{\operatorname{\sf ind}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Diff}{\operatorname{Diff}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}



\newcommand{\inbfpare}[1]{{%
  \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % 
  \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}%
}}
\newcommand{\comment}[1]{{}}

\def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}}
\def\endproof{\blacksquare}
\def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}}


\makeatletter

%\@ifundefined{Bbb}
%     {\newcommand{\Bbb}[1]{{\mathbb #1}}}%
%{}%     {\edef\Bbb#1{{\Bbb #1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Pagestyle                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
\newcommand{\ps@verbit}{%
  \renewcommand{\@oddhead}{%
          \scriptsize {\it \small Hodge theory, lecture 4 \hfil
  \tiny M. Verbitsky }}
  \renewcommand{\@evenhead}{\@oddhead}
  \renewcommand{\@oddfoot}{\hfil\thepage\hfil}
  \renewcommand{\@evenfoot}{\@oddfoot}}
 
\pagestyle{verbit}


   \setlength\paperheight {10in}%
    \setlength\paperwidth  {13.5in}
\setlength{\textwidth}{0.8\paperwidth}
\setlength{\textheight}{0.8\paperheight}

 \setlength{\pdfpageheight}{\paperheight}
 \setlength{\pdfpagewidth}{\paperwidth}
\addtolength{\topmargin}{-20mm}
\addtolength{\leftmargin}{-25mm}
\addtolength{\rightmargin}{-25mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Lemma, sublemma, corollary, proposition, theorem,             %
% definition,example defined there:                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{section}
\newcounter{Mycounter}[section]
\newcounter{lemma}[section]
\setcounter{lemma}{0}
\renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}}
\newcommand{\lemma}{%
     \setcounter{lemma}{\value{Mycounter}}
     \refstepcounter{lemma}
     \stepcounter{Mycounter}
     {\bf \green LEMMA:\ }}

\newcounter{claim}[section]
\setcounter{claim}{0}
\renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}}
\newcommand{\claim}{%
     \setcounter{claim}{\value{Mycounter}}
     \refstepcounter{claim}
     \stepcounter{Mycounter}
     {\bf \green CLAIM:\ }}

\newcounter{corollary}[section]
\setcounter{corollary}{0}
\renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}}
\newcommand{\corollary}{%
     \setcounter{corollary}{\value{Mycounter}}
     \refstepcounter{corollary}
     \stepcounter{Mycounter}
     {\bf \green COROLLARY:\ }}

\newcounter{theorem}[section]
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}}
\newcommand{\theorem}{%
     \setcounter{theorem}{\value{Mycounter}}
     \refstepcounter{theorem}
     \stepcounter{Mycounter}
     {\bf \green THEOREM:\ }}

\newcounter{conjecture}[section]
\setcounter{conjecture}{0}
\renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}}
\newcommand{\conjecture}{%
     \setcounter{conjecture}{\value{Mycounter}}
     \refstepcounter{conjecture}
     \stepcounter{Mycounter}
     {\bf \green CONJECTURE:\ }}

\newcounter{proposition}[section]
\setcounter{proposition}{0}
\renewcommand{\theproposition}
       {\noindent{Proposition \thesection.\arabic{proposition}}}
\newcommand{\proposition}{%
     \setcounter{proposition}{\value{Mycounter}}
     \refstepcounter{proposition}
     \stepcounter{Mycounter}
     {\bf \green PROPOSITION:\ }}

\newcounter{definition}[section]
\setcounter{definition}{0}
\renewcommand{\thedefinition}
       {\noindent{Definition~\thesection.\arabic{definition}}}
\newcommand{\definition}{%
     \setcounter{definition}{\value{Mycounter}}
     \refstepcounter{definition}
     \stepcounter{Mycounter}
     {\bf \green DEFINITION:\ }}


\newcounter{example}[section]
\setcounter{example}{0}
\renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}}
\newcommand{\example}{%
     \setcounter{example}{\value{Mycounter}}
     \refstepcounter{example}
     \stepcounter{Mycounter}
     {\bf \green EXAMPLE:\ }}

\newcounter{remark}[section]
\setcounter{remark}{0}
\renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}}
\newcommand{\remark}{%
     \setcounter{remark}{\value{Mycounter}}
     \refstepcounter{remark}
     \stepcounter{Mycounter}
     {\bf \green REMARK:\ }}


\newcounter{observation}[section]
\setcounter{observation}{0}
\renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}}
\newcommand{\observation}{%
     \setcounter{observation}{\value{Mycounter}}
     \refstepcounter{observation}
     \stepcounter{Mycounter}
     {\bf \green OBSERVATION:\ }}

\newcounter{question}[section]
\setcounter{question}{0}
\renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}}
\newcommand{\question}{%
     \setcounter{question}{\value{Mycounter}}
     \refstepcounter{question}
     \stepcounter{Mycounter}
     {\bf \green QUESTION:\ }}

\newcommand{\exercise}{{\bf \green EXERCISE:\ }} 
\newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} 
\newcommand{\proof}{{\bf \green Proof:\ }} 


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small lecture 4: Sobolev $L^2$-spaces and Rellich lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, February 3, 2018 } 
\end{center}



\newpage

{\bf \blue Banach spaces}

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

\definition
A {\bf \blue Banach space} is a complete normed
vector space.

\theorem
{\bf \red A space of bounded continuous functions  on $M$
with $C^0$-topology is Banach.}

\proof
A uniform limit of continuous functions is continuous
(Weierstrass), and a limit of a Cauchy sequence of functions
in $C^0(M)$ exists pointwise because $\R$ is complete.
\endproof



\newpage

{\bf \blue Stone-Weierstrass approximation theorem}



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

\theorem ({\bf \blue Stone-Weierstrass approximation theorem})
Let $M$ be a compact manifold
and  $A\subset  C^0 M$ be a subring separating points,
and $\bar A$ its closure. {\bf \red Then  $\bar A= C^0M$.}

\proof Handouts or the next lecture.
\endproof




\newpage

{\bf \blue Hilbert spaces (reminder)}

\definition
{\bf \blue Hilbert space} is a complete, infinite-dimensional
Hermitian space which is second countable
(that is, has a countable dense set).

\definition
    {\bf \blue Orthonormal basis} in a Hilbert space $H$
    is a set of pairwise orthogonal vectors $\{x_\alpha\}$
    which satisfy $|x_\alpha|=1$, and such that $H$ is the
    closure of the subspace generated by the set $\{x_\alpha\}$.

\theorem
{\bf \red Any Hilbert space has a basis, and all such bases are countable.}

\theorem
    {\bf \red All Hilbert spaces are isometric}.

    \proof Each Hilbert space has a countable orthonormal basis.
    \endproof

\newpage

{\bf \blue Fourier series}

\claim {\bf\blue ("Fourier series")}
Functions $e_k(t)= e^{ 2\pi\1 k t}$, $k\in \Z$ on $S^1=\R/\Z$
{\bf \red form an orthonormal basis in the space $L^2(S^1)$}
of square-integrable functions on the circle.


\proof
Orthogonality is clear from
$\int_{S^1} e^{ 2\pi\1 k t} dt =0$ for all $k\neq 0$ (prove it).
To show that the space of Fourier polynomials $\sum_{i=-n}^n a_k e_k(t)$
is dense in the space of continuous
functions on circle, use the Stone-Weierstrass approximation theorem,
applied to the ring $R= \langle \sin(mx), \cos (nx)\rangle$
of functions obtained from real 
and imaginary parts of $e^{ 2\pi\1 k t}$.
\endproof

\definition
{\bf \blue Fourier monomials}
on a torus are functions $F_{l_1, ..., l_n}:= \exp(2\pi\1\sum_{i=1}^n l_it_i)$,
where $l_1, ..., l_n\in \Z$. 

\claim
Fourier monomials {\bf \red form an orthonormal basis in the space 
$L^2(T^n)$} of square-integrable functions on the torus $T^n$.

\proof The same. \endproof


\newpage

{\bf \blue $L^2$-norms on vector spaces}

\theorem
Let $V$ be a vector space, and $g_1$, $g_2$
two scalar products. We say that
{\bf \blue $g_1$ is bounded by $g_2$}
if for some $C>0$, one has $g_1 \leq C g_2$.

\exercise
Prove that {\bf \purple this is equivalent to the continuity
of the map $(V, g_2)\arrow (V, g_1)$.} 

\remark
Let $g_1$ be bounded by $g_2$.
{\bf \purple Then the identity map extends to
a continuous map on the corresponding completion spaces
  $L^2(V, g_2) \arrow L^2(V, g_1)$.}

\remark
The topology induced by $g_1$ {\bf \purple is equivalent to topology
induced by $g_2$ if and only if
 $C^{-1} g_2 \leq g_1 \leq C g_2$.}

\newpage

{\bf \blue Sobolev's $L^2$-norm on $C^\infty_c(\R^n)$}

\definition
Denote by $C^\infty_c(\R^n)$ the space of smooth functions
with compact support. For each differential monomial
\[
   P_\alpha = \frac
   {\partial ^{k_1}}{\partial x_1^{k_1}}\frac
   {\partial ^{k_2}}{\partial x_2^{k_2}}...
   \frac{\partial ^{k_n}}{\partial x_1^{k_n}}
\]
consider the corresponding partial derivative
\[
P_\alpha(f) = \frac
   {\partial ^{k_1}}{\partial x_1^{k_1}}
   \frac{\partial ^{k_2}}{\partial x_2^{k_2}}... 
   \frac{\partial ^{k_n}}{\partial x_1^{k_n}}f.
\]
Given $f\in C^\infty_c(\R^n)$, one defines   
{\bf \blue the $L^2_p$ Sobolev's norm $|f|_{p}$}
as follows:
\[
|f|_s^2= \sum_{\deg P_\alpha\leq p} 
 \int \left|P_\alpha(f) \right |^2\Vol
\]
where the sum is taken over all differential
monomials $P_\alpha$ of degree $\leq p$,
and  $\Vol= dx_1\wedge dx_2\wedge ... dx_n$ -
the standard volume form. 

\remark Same formula defines {\bf \blue Sobolev's $L^2$-norm
$L^2_p$ on the space of smooth functions on a torus $T^n$.}

\newpage

{\bf \blue Sobolev's $L^2$-norm on a torus}

\claim
The Fourier monomials
$F_{l_1, ..., l_n}:= e^{2\pi\1\sum l_it_i}$
{\bf \red are eigenvectors for the differential monomials 
$P_\alpha = \frac
   {\partial ^{k_1}}{\partial x_1^{k_1}}\frac
   {\partial ^{k_2}}{\partial x_2^{k_2}}...
   \frac{\partial ^{k_n}}{\partial x_1^{k_n}}$.
Moreover, $P_\alpha(F_{l_1, ..., l_n})=
\prod_{i=1}^n (2\pi\1 k_i)^{l_i}$.}
\endproof

\corollary
The Fourier monomials are orthogonal in the Sobolev's
$L^2_p$-metric, and 
\[ 
  |F_{l_1, ..., l_n}|^2_{2,p}=
  \sum_{k_1+ ...+ k_n=1}^p \prod_{i=1}^n (2\pi l_i)^{2k_i}.
\]
\endproof


\newpage

{\bf \blue Weak convergence (reminder)}

\definition
Let $x_i \in H$ be a sequence of points in a Hilbert space $H$.
We say that $x_i$ {\bf \blue weakly converges} to $x\in H$
if for any $z\in H$ one has $\lim_i g(x_i, z) = g(x, z)$.

\remark 
Let $y(i)= \alpha_j(i) e_j$ be a sequence of points in 
a a Hilbert space with orthonormal basis $e_i$. {\bf
  \purple Then
$y(i)$ converges to $y= \sum_j \alpha_j e_j $ if and
only if $\lim_i \alpha_j(i)=\alpha_i$.}

\claim
For any sequence $\{y(i)=\sum_j \alpha_j(i) e_j\}$ of points in a unit ball,
{\bf \red there exists a subsequence $\{\tilde y(i)=\tilde \alpha_j(i) e_i\}$
weakly converging to $y\in H$.}

\proof Indeed, $|\alpha_j(i)|\leq 1$, hence
there exist a subsequence  $\tilde y(i)= \tilde \alpha_j(i) x_j$
with $\tilde \alpha_j(i)$ converging for each $j$. The limit belongs to the
unit ball because otherwise $\left|\sum_{j=1}^n \tilde \alpha_j(i) e_j\right|>1$,
which is impossible.
\endproof

\remark
Note that {\bf \red the function $x\arrow |x |$ 
is not continuous in weak topology.}
Indeed, weak limit of $\{e_i\}$ is 0. The proof above shows that
$|\cdot|$ is semicontinuous.


\newpage

{\bf \blue Compact operators (reminder)}

\definition
{\bf \blue Precompact set} is a set which has compact closure.
{\bf \blue A compact operator} is an operator which maps bounded 
sets to precompact.

\theorem
Let $A:\; H \arrow H_1$ be an operator on Hilbert spaces.
Then {\bf \red $A$ is compact if and only if it maps weakly 
convergent sequences to convergent ones.}


\newpage

{\bf \blue Rellich lemma for a torus}

\theorem {\bf \blue (Rellich lemma for a torus)}\\
{\bf \red The identity map $L^2_p(T^n)\arrow L^2_{p-1}(T^n)$.
is compact.} 

\pstep Consider, instead of $L^2_p$-metric,
the metric $q_p$ which is orthogonal
in the same basis and satisfies 
$|F_{l_1, ..., l_n}|_{q_p}:= 1+ (2\pi)^p\sum_{i=1}^n l_i^{p}$.
Clearly, $|F_{l_1, ..., l_n}|_{q_p} \leq |F_{l_1, ..., l_n}|_{2,p}$
and $|F_{l_1, ..., l_n}|_{q_p}\geq C^{-1} |F_{l_1, ..., l_n}|_{2,p}$,
where $C$ is a number of differential monomials of degree $p$.
Therefore, {\bf \purple $q_p$ and $L^2_p$ induce the same topology,} and
it would suffice to prove the Rellich lemma for 
the identity map $L^2(T^n, q_p)\arrow L^2(T^n, q_{p-1})$.

{\bf \green Step 2:} Now, 
\[
\frac{|F_{l_1, ..., l_n}|^2_{q_p}}{|F_{l_1, ..., l_n}|^2_{q_{p-1}}}=
\frac{\sum_{i=1}^n (2\pi)^p l_i^{2p}}{\sum_{i=1}^n (2\pi)^{p-1} l_i^{2p-2}} 
\geq \frac {n}{\max l_i^2}.
\]

{\bf \green Step 3:}
Let $x_i\in L^2(T^n, q_p)$ be weakly converging to $x$, with $|x_i|_{q_p}< 1$.
%This means that  $x_i = \sum \alpha_{l_1, ..., l_n}(i) F_{l_1, ..., l_n}$
%and $\lim_i  \alpha_{l_1, ..., l_n}(i) =  \alpha_{l_1, ..., l_n}$.
Let $x_i = y_i + z_i$, with $y_i$ being the sum of
all Fourier terms with $\max |l_i| < N$, and $z_i$ the rest.
Then $|z_i-z|_{q_{p-1}} < \frac {\sqrt n} N |z_i-z|_{q_{p}} <
\frac {2\sqrt n} N $,
and $y_i$ converges to $y$ because it is a sum of finitely
many terms which all converge. {\bf \purple
We obtain that $\lim_i |x_i -x|_{q_{p-1}}=0$,
hence a $x_i$ (strongly) converges to $x$.}
\endproof



\newpage

{\bf \blue Franz Rellich (1906-1955)}

\centerline{\epsfig{file=Rellich.jpg,width=0.25\linewidth}}

{\small      After Weyl's resignation [from G\"ottingen], his former
     assistant, Franz Rellich, became Institute Director
     ... Rellich had only a low-level appointment and
     ... was not an established figure ... There was need
     for a prominent mathematical figure who was suitable
     politically to take over the leadership in
     Gottingen. Furthermore, in mid-December, Rellich was
     ordered to report on January 7 for ten weeks to a
     field-sports camp near Berlin. This was, in fact, a
     mistake, since Rellich, as an Austrian citizen, was
     not subject to such forced training regimens. When he
     arrived at the camp, he was not admitted on these
     grounds. However, on December 27, the Curator had,
     after some hesitation, replaced Rellich with Werner
     Weber as acting director of the Mathematical
     Institute. Rellich himself would lose his position at
     Gottingen six months later, on June 18.
-- S. L. Segal, {\it Mathematicians under the Nazis}}


\newpage

{\bf \blue Rellich lemma for $C^\infty_K(\R^n)$}

\corollary
Let $C^\infty_K(\R^n)$ be the space of smooth functions
on $\R^n$ with support in a compact set $K$. {\bf \red Then the identity
map \[ L^2_p(C^\infty_K(\R^n))\arrow L^2_{p-1}(C^\infty_K(\R^n))\]
is compact.}

\proof
We consider a quotient map $\R^n \arrow T^n$ which is
bijective on $K$ for an appropriate choice of a lattice. 
This embeds $C^\infty_K(\R^n)$
to $C^\infty(T^n)$, and this embedding
is compatible with the $L^2_p$-norms.
\endproof


\newpage

{\bf \blue Sobolev's $L^2$-norm on a compact manifold}

\definition
Let $M$ be a manifold, $\{U_i\}$ a finite atlas,
and $\{\psi_i\}$ the corresponding partition of unity.
We will identify $U_i$ with bounded subsets in $\R^n$.
Given a function $f\in C^\infty(M)$, define
{\bf \blue the Sobolev $L^2_p$-metric}
$|f|_{2,p}^2$ as $\sum |f\psi_i|^2_{2,p}$,
where $ f\psi_i$ is considered as a function 
with compact support on $U_i \subset \R^n$,
and $\cdot|_{2,p}$ is the Sobolev $L^2_p$-metric
on $C^\infty_c(\R^n)$.

\proposition
The topology induced on $C^\infty(M)$ 
by $L^2_p$ {\bf \red is independent from the
choice of $\{U_i\}$ and $\{\psi_i\}$.}

\proof
Let $\Psi:\; \R^n \arrow \R^n$ be a map with
uniformly bounded partial derivatives up to $p$-th.
From the definition of the $L^2_p$-norm and the chain
rule it follows that 
\[ C^{-1} |f|_{2,p}^q \leq |\Psi^* f|_{2,p}^q \leq C |f|_{2,p}^q 
\]
where the constant $C$ depends on the supremum of
partial derivatives of $\Psi$. Then, for any refinement
$\{V_j\}$ of $\{U_i\}$ and the corresponding partition
of unity $\{\phi_j\}$, the $L^2_p$-norm of $f\psi_i$
associated with $\{V_j, \phi_j\}$ is bounded by the one associated
with $\{U_i, \psi_i\}$. For the same reason
the $L^2_p$-norm of $f\phi_j$ associated with
$\{V_j, \phi_j\}$ is bounded by the one  associated
with $\{U_i, \psi_i\}$. This gives 
an estimate of form $C^{-1} g_2 \leq g_1 \leq C g_2$
for $L^2_p$-metrics associated with a cover and its refinement.
To obtain a similar estimate for two different
covers, we find a common refinement.
\endproof


\newpage

{\bf \blue Rellich lemma for $C^\infty(M)$.}

\theorem {\bf \blue (Rellich lemma)}
Let $M$ be a compact manifold. {\bf \red Then
the identity map $L^2_p(M)\arrow L^2_{p-1}(M)$
is compact. }

\pstep
Let  $\{U_i\}$ be a finite atlas on $M$
and $\{\psi_i\}$ the corresponding partition of unity.
We will identify $U_i$ with bounded subsets in $\R^n$.
Then $|f|_{2,p}^2= \sum_i |\psi_i f|_{2,p}^2$,
where the second $|\cdot|_{2,p}^2$-norm is
taken on a bounded subset in $\R^n$.

{\bf \green Step 2:} Let $f_j\in L^2_p(M)$ be a sequence weakly
converging to $f$. Then $\psi_i f_j$ weakly converges to
a function $\tilde f_i$ with support in $\Supp(\psi_i)$.
Using Rellich lemma for functions on $\R^n$ with compact support,
we obtain that $\psi_i f_j$ converges in $L^2_{p-1}$ to  $\tilde f_i$.
Then $f_j = \sum_i \psi_i f_j$ converges in $L^2_{p-1}$ to 
$\sum_i \tilde f_i$.
\endproof




\end{document}
