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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: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\proof}{{\bf \green Proof:\ }} \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{Observation \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{exercise}[section] \setcounter{exercise}{0} \renewcommand{\theexercise}{\noindent{Exercise \thesection.\arabic{exercise}}} \newcommand{\exercise}{% \setcounter{exercise}{\value{Mycounter}} \refstepcounter{exercise} \stepcounter{Mycounter} {\bf \green EXERCISE:\ }} \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:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Locally conformally K\"ahler manifolds \\[15mm] \small lecture 9: holomorphic contractions and Riesz-Schauder theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] April 14, 2014 } \end{center} \newpage {\bf\blue LCK manifolds (reminder)} \definition Let $(M,I, \omega)$ be a Hermitian manifold, $\dim_\C M >1$. Then $M$ is called {\bf \blue locally conformally K\"ahler} (LCK) if $d\omega=\omega\wedge\theta$, where $\theta$ is a closed 1-form, called {\bf \blue the Lee form}. \definition {\bf \blue A manifold is locally conformally K\"ahler} iff it admits a K\"ahler form taking values in a positive, flat vector bundle $L$, called {\bf \blue the weight bundle}. \definition {\bf \blue Deck transform}, or {\bf \blue monodromy maps} of a covering $\tilde M \arrow M$ are elements of the group $\Aut_M(\tilde M)$. {\bf \purple When $\tilde M$ is a universal cover, one has $\Aut_M(\tilde M)=\pi_1(M)$.} \definition {\bf \blue An LCK manifold} is a complex manifold such that its universal cover $\tilde M$ is equipped with a K\"ahler form $\tilde \omega$, and the deck transform acts on $\tilde M$ by K\"ahler homotheties. \theorem {\bf \red These three definitions are equivalent}. \newpage {\bf \blue Conical K\"ahler manifolds (reminder)} \definition Let $(X,g)$ be a Riemannian manifold, and $C(X):= X \times \R^{>0}$, with the metric $t^2 g+ dt^2$, where $t$ is a coordinate on $\R^{>0}$. Then $C(X)$ is called {\bf \blue Riemannian cone} of $X$. {\bf \purple Multiplicative group $\R^{>0}$ acts on $C(X)$ by homotheties, $(m, t) \arrow (m, \lambda t)$.} \definition Let $(X,g)$ be a Riemannian manifold, $C(X):= X \times \R^{>0}$ its Riemannian cone, and $h_\lambda$ the homothety action. Assume that $(X,g)$ is equipped with a complex structure, in such a way that $g$ is K\"ahler, and $h_\lambda$ acts holomorphically. Then $C(X)$ is called {\bf \blue a conical K\"ahler manifold}. In this situation, $X$ is called {\bf \blue Sasakian manifold}. \remark A {\bf \blue contact manifold} is defined as a manifold $X$ with symplectic structure on $C(X)$, and $h_\lambda$ acting by homotheties. In particular, {\bf \purple Sasakian manifolds are contact}. {\bf \green Sasakian geometry is an odd-dimensional counterpart to K\"ahler geometry} \example Let $L$ be a positive holomorphic line bundle on a projective manifold. {\bf \purple Then the total space of its unit $S^1$-fibration is Sasakian.} \newpage {\bf\blue K\"ahler potentials and plurisubharmonic functions (reminder)} \definition A real-valued smooth function on a complex manifold is called {\bf\blue plurisubharmonic (psh)} if the (1,1)-form $dd^c f$ is positive, and {\bf\blue strictly plurisubharmonic} if $dd^c f$ is an Hermitian form. \remark Since $dd^c f$ is always closed, {\bf \purple it is also K\"ahler when it is strictly positive.} \definition Let $(M,I,\omega)$ be a K\"ahler manifold. {\bf \blue K\"ahler potential} is a function $f$ such that $dd^c f=\omega$. \theorem Let $S$ be a Sasakian manifold, $C(S)=S \times \R^{>0}$ its cone, $t$ the coordinate along the second variable, and $r=t\frac{d}{dt}$. {\bf \red Then $t^2$ is a K\"ahler potential on $C(S)$.} \newpage {\bf\blue LCK manifolds with potential (reminder)} \definition Let $M$ be an LCK manifold, and $(\tilde M, \tilde \omega)$ its K\"ahler covering. It is called {\bf \blue LCK manifold with potential} if $\tilde M$ admits an automorphic K\"ahler potential $\phi:\; \tilde M \arrow \R^{>0}$, $dd^c \phi=\tilde \omega$, which is {\bf \blue proper} (preimage of a compact is again compact). \theorem {\bf \purple The property of being LCK with potential is stable under small deformations}. \remark For any complex submanifold $Z\subset M$ of an LCK manifold with potential, $Z$ is also an LCK manifold with potential. \theorem Let $M$ be an LCK manifold, $\Gamma\subset \R^{>0}$ the monodromy group, and $(\tilde M, \tilde \omega)$ its K\"ahler covering, with $\tilde M/\Gamma=M$. Assume that $\tilde \omega$ admits a $\Gamma$-automorphic K\"ahler potential $\phi$. {\bf \red The map $\phi$ is proper if and only if $\Gamma=\Z$.} \theorem Let $M$ be an LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. Then a metric completion $\tilde M_c$ {\bf \red admits a structure of a complex manifold,} compatible with the complex structure on $\tilde M \subset\tilde M_c$. Moreover, the monodromy action on $\tilde M$ is extended to a holomorphic automorphism of $\tilde M_c$. \newpage {\bf \blue Normal families of functions (reminder)} \definition Let $C(M)$ be the space of functions on a topological space. {\bf \blue Topology of uniform convergence on compacts} (also known as {\bf \blue compact-open topology}; usually denoted as $C^0$) is a topology on $C(M)$ where a base of open sets is given by \[ U(X, C):= \{f\in C(M)\ \ |\ \ \sup_K |f|< C\}, \] for all compacts $K\subset M$ and $C>0$. A sequence $f_i$ of functions converges to $f$ if it converges to $f$ uniformly on all compacts. \definition Let $M$ be a complex manifold, and ${\cal F}$ a family of holomorphic functions $f_i \in H^0(\calo_M)$. We call ${\cal F}$ {\bf \blue a normal family} if for each compact $K\subset M$ there exists $C_K>0$ such that for each $f\in {\cal F}$, $\sup_K |f| \leq C_K$. \theorem Let $M$ be a complex manifold and ${\cal F}\subset H^0(\calo_M)$ a normal family of functions. Denote by $\bar{\cal F}$ its closure in $C^0$-topology. {\bf \red Then $\bar{\cal F}$ is compact and contained in $H^0(\calo_M)$.} \newpage {\bf \blue Banach space of holomorphic functions} \definition A {\bf\blue Banach space} is a complete normed vector space. \theorem Let $M$ be a complex manifold, and $H^0_b(\calo_M)$ the space of all bounded holomorphic functions, equipped with {\bf \blue the sup-norm} $|f|_\sup:=\sup_M |f|$. {\bf \red Then it is a Banach space.} {\bf \green Proof. Step 1:} Let $\{f_i\}\in H^0_b(\calo_M)$ be a Cauchy sequence in $\sup$-norm. {\bf \purple Then $\{f_i\}$ converges to a continuous function $f$} in $\sup$-topology. {\bf \green Step 2:} Since $\{f_i\}$ is a normal family, it has a subsequence which converges in $C^0$-topology to $\tilde f\in H^0(\calo_M)$. However, the {\bf \purple $C^0$-topology is weaker than the $\sup$-topology, hence $\tilde f=f$.} Therefore, $f$ is holomorphic. \endproof \newpage {\bf \blue Compact operators} \definition A {\bf \blue bounded subset} of a topological vector space $V$ is a set $B\subset V$ such that for any open neighbourhood $U \ni 0$, there exists $\lambda>0$ such that $\lambda B\subset U$ \remark Bounded subsets of normed spaces are subsets which are contained in a ball of a sufficiently big radius. \definition A subset of a topological space is called {\bf \blue precompact} if its closure is compact. \definition Let $V, W$ be topological vector spaces, and $\phi:\; V \arrow W$ a continuous linear operator. It is called {\bf \blue compact} if an image of any bounded set is precompact. \exercise Let $V=H^0(\calo_M)$ be a space of holomorphic functions on a complex manifold $M$ with $C^0$-topology. {\bf \purple Prove that any bounded subset of $V$ is precompact.} In this case, the identity map is a compact operator. \remark By Riesz theorem, {\bf \purple a closed ball in a normed vector space $V$ is never compact,} unless $V$ is finite-dimensional. This means that $(H^0(\calo_M), C^0)$ does not admit a norm. \newpage {\bf \blue Holomorphic contractions} \definition {\bf\blue Contraction} of a manifold $M$ to a point $x\subset M$ is a morphism $\phi:\; M \arrow M$ such that for any compact subset $K\subset M$ and any open $U\ni x$ there exists $N>0$ such that for all $n>N$, the map $\phi^n$ maps $K$ to $U$. \theorem Let $X$ be a complex variety, and $\gamma:\; X \arrow X$ a holomorphic contraction such that $\gamma(X)$ is precompact. Consider the Banach space $V=H^0_b(\calo_X)$ with sup-metric. {\bf \red Then $\gamma^*:\; V \arrow V$ is compact,} and {\bf \red its operator norm $\|\gamma^*\|:= \sup_{|v|\leq 1}|\gamma^*(v)|$ is strictly less than 1. } {\bf \green Proof. Step 1:} Let $B_C:=\{ v\in V\ \ | \ \ |v|_\sup \leq C\}$. Then \[|\gamma^* f|_\sup= \sup_{x\in \overline{\gamma(X)}} |f(x)|. \] Therefore, {\bf \purple for any sequence $\{f_i\}$ converging in $C^0$-topology, the sequence $\{\gamma^* f_i\}$ converges in $\sup$-topology.} However, $B_C$ is precompact in $C^0$-topology, because it is a normal family. Then $\gamma^* B_C$ is precompact in $\sup$-topology. \newpage {\bf \blue Holomorphic contractions (part 2)} \exercise Prove {\bf \blue the maximum principle}: {\bf \red a non-constant holomorphic function cannot have any non-strict maxima.} {\bf \green Step 2:} Since $\sup_X |\gamma^* f|= \sup_{\gamma(X)} |f| \leq \sup_X |f|$, one has $\|\gamma^*\|\leq 1$. If this inequality is not strict, for some sequence $f_i\in B_1$ one has $\lim_i \sup_{x\in \gamma(X)} |f_i(x)|=1$. Since $B_1$ is a normal family, $f_i$ has a subsequence converging in $C^0$-topology to $f$. Then $\gamma(f_i)$ converges to $\gamma(f)$ in $\sup$-topology, giving $\lim_i \sup_{x\in \gamma(X)} |f_i(x)|= \sup_{x\in \gamma(X)} |f(x)|=1$. {\bf \purple Since a holomorphic functions has no strict maxima, this means that $|f(x)| >1$ somewhere on $X$.} Then $f$ cannot be a $C^0$-limit of $f_i\in B_1$. \endproof %\newpage % %{\bf \blue Logarithm in Banach spaces} % %\definition %Let $A$ be a %Define its logarithm as a sum %\[ \log(A) := \sum_{i=1}^\infty \frac{(1-A)^i}{i}. %\] % % %\theorem %Let $\tilde M_c$ be a completion of a $\Z$-covering $\tilde M$ of %an LCK manifold with potential, and $\gamma:\; \tilde M_c\arrow \tilde M_c$ %the holomorphic contraction, associated with the %generator of the $\Z$-action. Consider the logarithm of $\gamma$ %defined as above. %{\bf \red %Then $\log\gamma^*$ converges in $C^0$-topology on $H^0(\tilde M_c)$.} %Moreover, $r:=\log\gamma^*$ satisfies the Leibnitz identity, and %$e^r=\gamma^*$. % %{\bf \green Proof:} %Convergency of a logarithm \newpage {\bf \blue Hopf manifolds and finite vectors} \definition Let $A\in \End(\C^n)$ be an invertible linear endomorphism with all eigenvalues $|\alpha_i| <1$. The quotient $H:= (\C^n \backslash 0)/\langle A\rangle$ is called {\bf \blue a linear Hopf manifold}. {\bf \green Theorem 1:} Let $M$ be an LCK manifold with potential, $\dim_\C M>2$. {\bf \red Then $M$ admits a holomorphic embedding to a linear Hopf manifold. } \definition Let $\gamma$ be an endomorphism of a vector space $V$. A vector $v\in V$ is called {\bf \blue $\gamma$-finite} if the space $\langle v, \gamma(v), \gamma^2(v), ... \rangle$ is finite-dimensional. {\bf \green Theorem 2:} Let $M$ be an LCK manifold with potential, $\dim_\C M>2$, and $\tilde M$ its K\"ahler $\Z$-covering. Consider a metric completion $\tilde M_c$ with its complex structure and a contraction $\gamma:\; \tilde M_c \arrow \tilde M_c$ generating the $\Z$-action. Let $H^0(\calo_{\tilde M_c})_f$ be the space of functions which are $\gamma^*$-finite. {\bf \red Then $H^0(\calo_{\tilde M_c})_f$ is dense in $\sup$-topology on each compact subset of $\tilde M_c$.} We deduce Theorem 1 from Theorem 2, and then prove Theorem 2 using Riesz-Schauder Theorem. \newpage {\bf \blue Embedding into Hopf manifolds} {\bf \green Theorem 2 $\Rightarrow$ Theorem 1. Step 1:}\\ Let $W\subset H^0(\calo_{\tilde M_c})_f$ be an $m$-dimensional $\gamma^*$-invariant subspace $W$ with basis $w_1, ..., w_m$. Then the following diagram is commutative: \begin{equation*} \begin{CD} \tilde M@>\Psi>> \C^m \\ @V{\gamma}VV @VV{\gamma^*}V \\ M@>\Psi >> \C^m \end{CD}, \end{equation*} where $\Psi(x)=(w_1(x), w_2(x), ..., w_m(x))$. Suppose that the map $\Psi$ associated with a given $W\subset H^0(\calo_{\tilde M_c})_f$ is injective. Then {\bf \purple the quotient map gives an embedding $\Psi:\; \tilde M/\Z \arrow (\C^m\backslash 0)/\gamma^*$;} all eigenvalues of $\gamma^*$ are $<1$ because its operator norm is $<1$. {\bf \green Step 2:} To find an appropriate $W\subset H^0(\calo_{\tilde M_c})_f$, choose a holomorphic embedding $\Psi_1:\; \tilde M_c\hookrightarrow \C^n$, which exists because $\tilde M_c$ is Stein. Let $\tilde w_1, ..., \tilde w_n$ be the coordinate functions of $\Psi_1$. Theorem 2 allows one to approximate $\tilde w_i$ by $w_i \in H^0(\calo_{\tilde M_c})_f$ in $C^0$-topology. {\bf \purple Choosing $w_i$ sufficiently close to $\tilde w_i$ in a compact fundamental domain of $\Z$-action, we obtain that $x\arrow (w_1(x), w_2(x), ..., w_n(x))$ is injective in a compact fundamental domain of $\Z$.} To finish the argument, take $W\subset H^0(\calo_{\tilde M_c})_f$ generated by the $\gamma^*$ from $w_1, ..., w_n$, and apply Step 1. \endproof \newpage {\bf \blue Riesz-Schauder Theorem} To find enough $\gamma^*$-finite vectors, we use the Riesz-Schauder Theorem. It is a Banach analogue of spectral theorem which easily follows from Fredholm theory. \theorem {\bf \blue (Riesz-Schauder)} \\ Let $F:\; V \arrow V$ be a compact operator on a Banach space. Then for each non-zero $\mu \in \C$, there exists a sufficiently big number $N\in \Z$ such that for each $n>N$ {\bf \red one has $V= \ker(F-\mu\Id)^n \oplus \overline{\im (F-\mu\Id)^n}$, where $\overline{\im (F-\mu\Id)^n}$ is closure of the image.} Moreover, $\ker(F-\mu\Id)^n$ is finite-dimensional and independent on $n$. %\exercise Deduce from the Riesz-Schauder % that {\bf \purple the same statement is true for any $F=P(K)$} %where $K$ is compact, and $P$ a polynomial, . \newpage {\bf \blue Riesz-Schauder Theorem and adic filtration} In our case the Riesz-Schauder theorem is especially effective. \def\m{{{\goth m}}} {\bf \green Proposition 1:} Fix a precompact subset $\tilde M^a_c:= \phi^{-1}([0, a[)$, where $\phi:\; \tilde M_c$ is the K\"ahler potential. Let $A$ be the ring of bounded holomorphic functions on $\tilde M^a_c$, and ${\goth m}$ the maximal ideal of the origin point. Clearly, $\gamma^*$ preserves $\m$ and all its powers. Let $P_k(t)$ be the minimal polynomial of $\gamma^*\restrict{A/\m^k}$. {\bf \red Then $\im(P_k(\gamma^*)\subset \m^k(A)$, and $\ker P_k(\gamma^*)$ generates $A/\m^k$.} {\bf \green Proof:} Since $P_k(t)$ is a minimal polynomial of $\gamma^*$ on $A/\m^k$, the endomorphism $P_k(\gamma^*)$ acts trivially on $A/\m^k$, hence it maps $A$ to $\m^k$. From Riesz-Schauder theorem applied to the Banach space $A$ and $F=P_k(\gamma^*)$ it follows that $A=\im(P_k(\gamma^*)+ \overline{\im (F-\mu\Id)^n}$ hence $\ker P_k(\gamma^*)$ generates $A/\m^k$. \endproof \newpage {\bf \blue $\m$-adic topology and $C^0$-topology} \definition Let $A$ be a ring and $\m$ an ideal. The base of open sets in {\bf \blue $\m$-adic topology on $A$} are $\m^k$ and their translates. Proposition 1 implies the following result \proposition Let $H^0(\calo_{\tilde M_c})_f\subset H^0(\calo_{\tilde M_c})$ be the set of $\gamma^*$-finite functions and $\m$ the maximal ideal of the origin in $\tilde M_c$. {\bf \red Then $H^0(\calo_{\tilde M_c})_f$ is dense in $\m$-adic topology.} \proof A subspace $V\subset A$ is dense in $\m$-adic topology in $A$ $\Leftrightarrow$ the quotient $V/v\cap \m^k$ surjects to $A/\m^k$. This is proven in Proposition 1 for the ring of bounded holomorphic functions on $\tilde M^a_c$. However, any such function can be extended to $\gamma^*$-finite function on $\tilde M_c$ using $\gamma^*$-action. \endproof Now Theorem 2 is implied by the following claim. \claim Let $X$ be a connected complex variety, $A$ the ring of bounded holomorphic functions, $x\in X$ a point, $\m\subset A$ its maximal ideal, and $R:\; A \arrow \hat A$ the natural map from $A$ to its $\m$-adic completion. {\bf \red Then $R$ is continuous in $C^0$-topology and induces homeomorphism of any bounded set to its image.} \newpage {\bf \blue $\m$-adic topology and $C^0$-topology, part 2} \claim Let $X$ be a connected complex variety, $A$ the ring of bounded holomorphic functions, $x\in X$ a point, $\m\subset A$ its maximal ideal, and $R:\; A \arrow \hat A$ the natural map from $A$ to its $\m$-adic completion. {\bf \red Then $R$ is continuous in $C^0$-topology and induces homeomorphism of any bounded set to its image.} {\bf \green Proof:} Continuity is clear because $C^0$-topology is equivalent to $C^1$-topology, $C^2$-topology and so on (Lecture 8). Therefore, taking successive derivatives in a point is continuous in $C^0$-topology. However, $R$ takes a function and replaces it by its Taylor series. To see that $R$ is a homeomorphism, notice that any bounded, closed subset of $A$ is compact, hence its image under a continuous map is also closed. \endproof \end{document}