
\documentclass{slides}

\usepackage{amssymb, amsmath, amscd, color, epsfig, units}
%\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}}
\renewcommand{\H}{{\Bbb H}}
\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}{{\it const}}
\newcommand{\capa}{{\sf capa}}
\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{\Vol}{\operatorname{Vol}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ham}{\operatorname{Ham}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Ann}{\operatorname{Ann}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Can}{\operatorname{Can}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\Symp}{\operatorname{Symp}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}

\newcommand{\coker}{\operatorname{coker}}
\newcommand{\slope}{\operatorname{slope}}
\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}}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Flux}{\operatorname{Flux}}
\newcommand{\Hess}{\operatorname{Hess}}

\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 Symplectic geometry, lecture 7 \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{\proof}{%
     {\bf \green Proof:\ }}
\newcommand{\pstep}{%
     {\bf \green Proof. Step 1:\ }}


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 7: Ekeland-Hofer theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
September 25, 2021
}
\end{center}

\newpage

{\bf \blue Gromov capacity (reminder)}

\definition
{\bf \green An (open) symplectic embedding} 
is an open embedding of symplectic manifolds,
symplectimorphic to its image. 

\definition
Let $(M,\omega)$ be a symplectic manifold,
and $r$ a supremum of radii of all symplectic
balls of the same dimension, admitting
a symplectic embedding to $M$.
The number $\capa(M,\omega):=\pi r^2$
is called {\bf \blue Gromov symplectic capacity}
of $M$.

\theorem {\bf \blue (Ekeland-Hofer)}\\
Let $\phi$ be an oriented diffeomorphism of symplectic manifold.
Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$
preserves the symplectic capacity of all open subsets.}

\proof Later today. \endproof

\newpage

{\bf \blue Ekeland-Hofer theorem (reminder)}


\theorem {\bf \blue (Ekeland-Hofer)}\\
Let $\phi$ be an oriented diffeomorphism of symplectic manifold.
Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$
preserves the symplectic capacity of all open subsets.}


Today, the Ekeland-Hofer theorem  {\bf \green will be deduced
from its linear version.}

\theorem {\bf \blue Ekeland-Hofer, the linear version}\\
Let $(V=\R^{2n}, \omega= \sum_i dp_i \wedge dq_i)$
be a symplectic vector space, and  $\phi:\; V \arrow V$
an oriented linear map which preserves the Gromov capacity
of all ellipsoids. {\bf \red Then $\phi$ is a symplectomorphism.}

\proof  Next lecture. \endproof

\newpage

{\bf \blue Hausdorff metric (reminder)}


\definition
Let $Z\subset M$ be a metric space.
{\bf \blue An $\epsilon$-neighbourhood}
$Z(\epsilon)$ of $Z$ is a union of all $\epsilon$-balls
centered in $Z$. {\bf \blue Hausdorff metric} $d_H$
 on closed subsets $M$ is
defined as follows: $d_H(X,Y)$ is infimum of all $\epsilon$
such that $Y\subset X(\epsilon)$ and $X\subset Y(\epsilon)$.

{\bf \green Properties of Hausdorff metric:}

1. Let $M$ be a metrizable topological space, 
and ${\cal C}$ the set of compact subsets of $M$.
Then the topology induced by the Hausdorff metric on ${\cal C}$
is independent from the choice of the metric on $M$.
{\bf \purple (Prove it!)}

2. Let $\phi_i:\; M \arrow N$ be a sequence
of continuous maps, and $\Gamma_{\phi_i}\subset M\times N$
their graphs. Suppose that $M, N$ are compact.
Then the sequence $\{\phi_i\}$ converges
to $\phi:\; M \arrow N$ (in the compact-open topology)
{\bf \purple if and only if $\Gamma_{\phi_i}$ converges to $\Gamma_\phi$
in the Hausdorff topology.}

\remark
Further on, we will consider
``the Hausdorff metric'' on the set of open
subsets of $M$ with compact closure.
{\bf \red This is in fact a pseudometric:}
two open subsets $U, V$ with the same closure
satisfy $d_H(U, V)=0$.

\newpage

{\bf \blue  Hausdorff distance and boundary}


%\lemma
%Let $E\subset \R^n$ be a convex set.
%{\bf \red Then $E(\epsilon)$ is convex. }
%
%\proof
%Given an interval $a_1, a_2\in E(\epsilon)$,
%take $b_1, b_2 \in E$ such that $d(a_i, b_i)< \epsilon$.
%Then the distance between any point of
%$[a_1, a_2]$ and the interval $[b_1, b_2]\subset E$
%is less than $\epsilon$, hence
%$[a_1, a_2]\subset E(\epsilon)$.
%\endproof

\lemma
Let $E_1, E_2$ be bounded open convex subsets in $\R^n$
and $U_i := \R^n \backslash E_i$. Then 
$d_H(E_1, E_2) \geq d_H(U_1, U_2)$.

\proof 
Suppose that $d_H(U_1, U_2)\geq \epsilon$.
Let $x\in U_1$ such that
$x\notin U_2(\epsilon)$. This means
that $d(x, U_2) \geq \epsilon$,
hence $B_\epsilon(x) \supset E_2$.
By Hahn-Banach separation theorem,
there exists a hyperplane $H$ passing through
$x$ such that all $E_1$ lies to one side
of this hyperplane. Since the distance
from the farthest point of 
$B_\epsilon(x)$ to this hyperplane is
$\epsilon$, and $B_\epsilon(x)\subset E_2$,
this implies that $d_H(E_1, E_2)\geq \epsilon$.
\endproof


\lemma
Let $E_1, E_2$ be bounded open convex subsets in $\R^n$
and $\6(E_1):= \bar E_i \backslash E_i$ denote {\bf \blue
the boundary} of $E_i$.
{\bf \red Then $d_H(E_1, E_2) \geq  d_H(\6 E_1, \6 E_2)$.}

\proof
Whenever $d_H(E_1, E_2)< \epsilon$,
we also have $d_H(U_1, U_2)< \epsilon$, where
$U_i = \R^n \backslash E_i$, hence
every point $x$ on a boundary of $E_1$
satisfies $x\in E_2(\epsilon) \cap U_2(\epsilon) =\6 E_2 (\epsilon)$.
\endproof

\newpage

{\bf \blue  Hausdorff distance between convex sets and homothety}


{\bf \green Claim 1:}
Let $E\subset \R^n$ be a convex set containing 0, and $\epsilon >0$.
Define $\lambda U:= \{x\in \R^{2n}\ \ | \ \  \lambda^{-1}x \in U\}$.
{\bf \red Then for each $\epsilon>0$ there exists $\delta>0$
such that any convex $E_1$ satisfying $d_H(E, E_1) < \epsilon$ also satisfies
$(1-\delta) E\subset E_1\subset (1+\delta) E$.}

\pstep
Define $u(\delta):\;\R^{>0} \arrow \R^{>0}$  as
\[ u(\delta):= 
\inf_{x\in  \6 E}\min [d(x, \6 (1-\delta) E), d(x, \6
  (1+\delta) E)].
\]
Since $\6 E$ is compact, and $d(x, \6 (1-\delta) E)$
is 1-Lipschitz as a function of $x$, 
the number $d(x, \6 (1-\delta) E)$
reaches its minumum somewhere on
$\6 E$, and $u(\delta)$ is positive.
An infimum of 1-Lipschitz functions
is 1-Lipschitz, hence $u$ is 1-Lipschitz.
{\bf \purple Therefore, there exists $\delta$ such that
$u(\delta) < \epsilon$.}


{\bf \green Step 2:} 
The inequality
$d_H(E, E_1) < \epsilon $ implies
$\6E_1 \subset \6E(\epsilon)$, by the previous
lemma.  Step 1 implies that {\bf \purple anything
which lies in an $\epsilon$-neighbourhood of $\6E$
belongs to the segment bounded by
$\6 (1-\delta)E$ and $\6(1-\delta) E)$.}
This implies that 
$(1-\delta) E\subset \6E_1\subset (1+\delta) E$.
Passing to convex hulls, we obtain
$(1-\delta) E\subset E_1\subset (1+\delta) E$.
\endproof


\newpage

{\bf \blue Symplectic capacity and Hausdorff convergence (remider)}

\claim
Let $U_i\subset \R^{2n}$ be a Cauchy sequence of 
bounded open subsets, containing 0, and $U$ their limit in the Hausdorff
metric. Assume that $U$ is convex.
{\bf \red Then $\lim_i\capa_G(U_i)=\capa_G(U)$,}
where $\capa_G(U)$ denotes the Gromov capacity.

\proof Let $\lambda U:= \{x\in \R^{2n}\ \ | \ \  \lambda^{-1}x \in U\}$.
By Claim 1, for any $\epsilon >0$, almost all elements of the sequence 
$U_i$ contain $(1-\epsilon)U$ and are contained in $(1+\epsilon)U$.
In this situation,
\[
\sqrt{1-\epsilon}\capa_G(U)\leq 
\capa_G(U_i)\leq \sqrt{1+\epsilon}\capa_G(U).
\]
\endproof

Now we can deduce Ekeland-Hofer theorem from
its linear version.

\theorem {\bf \blue Ekeland-Hofer, the linear version}\\
Let $(V=\R^{2n}, \omega= \sum_i dp_i \wedge dq_i)$
be a symplectic vector space, and  $\phi:\; V \arrow V$
an oriented linear map which preserves the Gromov capacity
of all ellipsoids. {\bf \red Then $\phi$ is a symplectomorphism.}


\newpage

{\bf \blue Image of an ellipsoid}

Recall that {\bf \blue $C^n$-topology} on 
functions $\phi:\; \R^n \arrow \R^k$ is the topology
of uniform convergence on compacts for the derivatives up
to $n$-th.

{\bf \green Claim 2:} 
Let $\phi_i:\; \R^n \arrow \R^n$ be a sequence
of diffeomorphisms converging to identity in $C^2$-topology,
and $E\subset \R^n$ an ellipsoid.
{\bf \red Then for $i$ sufficiently big, all $\phi_i(E)$ are 
convex.}

\proof
Let $u:\; \R^n \arrow \R$ be a quadratic function
$u(x_1, ..., x_n)= \sum x_i^2 a_i^2$ such
that $E= u^{-1}([0,1[)$, and 
$u_i:= (\phi_i^{-1})(u)$.
Then $\phi_i(E)= u^{-1}_i([0,1[)$.
Clearly, the $C^2$-convergence of
$\phi_i$ to identity implies
the $C^2$-convergence of
$u_i$ to $u$. The functions $u_i$ are convex for $i$
sufficiently big, because  {\bf \purple the Hessian $\Hess(u_i-u)$
uniformly converges to zero, and $\Hess(u)$
is positive definite.} On the other hand,
the preimage $u^{-1}_i([0,1[)$ is convex if
$u_i$ is convex.
\endproof



\newpage

{\bf \blue Ekeland-Hofer theorem deduced from its linear version (reminder)}


\theorem {\bf \blue (Ekeland-Hofer)}\\
Let $\phi$ be an oriented diffeomorphism of symplectic manifold.
Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$
preserves the symplectic capacity of all open subsets.}

\proof
Locally, every symplectic manifold is symplectomorphic
to a symplectic ball (Darboux). Therefore it would suffice
to prove the following (weaker) form of this theorem.

\theorem
Let $(B,\omega)\stackrel \phi \arrow (\R^{2n},\omega)$
be an open embedding, mapping a symplectic ball 
to $\R^{2n}$ with the usual symplectic structure,
mapping 0 to 0, and preserving the Gromov symplectic
capacities of all convex subsets. 
{\bf \red Then $\phi$ is a symplectomorphism}.

\pstep Let $\lambda>1$, and 
$\Gamma_\lambda:\; \R^n \arrow \R^n$
the homothety mapping $v$ to $\lambda v$.
By conformal invariance of $\capa_G$, 
the diffeomorphism
$\phi_\lambda:\; B \arrow \R^{2n}$,
defined as $\phi_\lambda(v):=\Gamma_\lambda(\phi(\Gamma_\lambda^{-1}(v)))$ 
preserves the  Gromov symplectic
capacities. {\bf \purple
If $\phi(x)= \sum_{i=1}^\infty P_i(x)$ is the
Taylor decomposition for $\phi$, with $P_i$
homogeneous polynomials of degree $i$, one has
$\phi_\lambda(x) = \sum_{i=1}^\infty \lambda^{i-1}P_i(x)$.}

\newpage

{\bf \blue Ekeland-Hofer theorem deduced from its linear version (2)}


{\bf \green Step 1:}
{\bf \purple Let  
$\phi_\lambda(v):=\Gamma_\lambda(\phi(\Gamma_\lambda^{-1}(v)))$.}
If $\phi(x)= \sum_{i=1}^\infty P_i(x)$ is the
Taylor decomposition for $\phi$, we have
$\phi_\lambda(x) = \sum_{i=1}^\infty \lambda^{i-1}P_i(x)$.

{\bf \green Step 2:}
For any diffeomorphism  $(B,\omega)\stackrel \phi \arrow (\R^{2n},\omega)$
and any ellipsoid $E\subset B$, {\bf \purple there exists
$\lambda_0>0$ such that $\phi_\lambda(E)$ is convex for any
 $\lambda>\lambda_0$.} Indeed, the second derivative of
$\phi_\lambda$ tends to 0 as $\lambda$ tends to infinity, 
hence for $\lambda$ sufficiently big this map 
maps $E$ to a convex set.

{\bf \green Step 3:}
In an open-compact topology,
$\lim\limits_{\lambda\rightarrow \infty}\phi_\lambda$
is equal to the differential  ${\goth D}:=D_0\phi$.
For each ellipsoid $E\subset B$, we have
${\goth D}(E)=\lim\limits_{\lambda\rightarrow \infty}\phi_\lambda(E)$
(in the Hausdorff topology). For $\lambda$ sufficiently
big, the set $\phi_\lambda(E)$ is convex by Claim 2, and on
convex subsets, the function $\capa_G$ is
continuous in the Hausdorff topology.
{\bf \purple This gives
\[ \capa_G({\goth D}(E))=\lim\limits_{\lambda\rightarrow \infty}
\capa_G(\phi_\lambda(E))=\capa_G(E),
\]
that is, ${\goth D}$ preserves the symplectic capacity.}

{\bf \green Step 4:} Using the linear Ekeland-Hofer, 
we obtain that ${\goth D}=D_0\phi$  is a symplectomorphism.
Since the choice of 0 was arbitrary, the same argument
proves that {\bf \purple the differential of $\phi$ preserves
the symplectic form everywhere  where it is defined.}
\endproof



\end{document}
