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\begin{document}

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\section*{Borel&#39;s theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statements}{Statements}\dotfill \pageref*{statements} \linebreak
\noindent\hyperlink{proof}{Proof}\dotfill \pageref*{proof} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Borel's Theorem} (also called \emph{Borel's Lemma}) says that every [[power series]] is the [[Taylor series]] of some [[smooth function]]. In other words: for every collection of prescribed [[partial derivatives]] at some point, there is a smooth function having these as its actual derivatives.

\hypertarget{statements}{}\subsection*{{Statements}}\label{statements}

\begin{theorem}
\label{basic}\hypertarget{basic}{}
The canonical map from the ring of germs of $C^\infty$ function at $0 \in \mathbb{R}^n$ to the ring of formal power series obtained by taking the Taylor series at $0$ is surjective.

\end{theorem}
There are many extensions and variants.

For $\mathbb{R}^{n+m}$ a [[Cartesian space]] of [[dimension]] $n+m \in \mathbb{N}$, write $C^\infty(\mathbb{R}^{n+m})$ for the $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth function]]s with values in $\mathbb{R}$.

Write $m^\infty_{\mathbb{R}^n \times \{0\}} \subset C^\infty(\mathbb{R}^{n+m})$ for the ideal of functions all whose [[partial derivatives]] along $\mathbb{R}^m$ vanish.

\begin{theorem}
\label{general}\hypertarget{general}{}
Forming the [[Taylor series]] constitutes an [[isomorphism]]

\begin{displaymath}
C^\infty(\mathbb{R}^{n+m})/m^\infty_{\mathbb{R}^n \times \{0\}}
  \stackrel{\simeq}{\longrightarrow}
  C^\infty(\mathbb{R}^n) [ [ Y_1, \cdots, Y_m] ]
\end{displaymath}
between smooth functions modulo those whose derivatives along $\mathbb{R}^m$ vanish and the ring of [[power series]] in $m$-variables over $C^\infty(\mathbb{R}^n)$.

\end{theorem}
This appears for instance as (\hyperlink{MoerdijkReyes}{Moerdijk-Reyes, theorem I.1.3}).

\hypertarget{proof}{}\subsection*{{Proof}}\label{proof}

There is a full proof in Moerdijk--Reyes, cited above. Here we prove Theorem \ref{basic} to indicate the method; the general version is not substantially different. (This is based on the proof in \href{https://en.wikipedia.org/wiki/Borel%27s_lemma}{the English Wikipedia article} at the time of writing, but with more details.)

\begin{proof}
A real [[power series]] at $0$ is given simply by an [[infinite sequence]] $c = (c_n)_{n\geq{0}}$ of [[real numbers]]. Given such a sequence, we would ideally use

\begin{displaymath}
f(x) = \sum_{n=0}^\infty c_n x^n ,
\end{displaymath}
but this is only correct if the sum [[convergence|converges]] on at least some [[neighbourhood]] of $0$ (in other words if the power series has a positive [[radius of convergence]]).

To ensure this, let $\psi$ be a [[smooth function|smooth]] [[bump function]] chosen so that $\psi(x) = 1$ for ${|x|} \leq 1$ and $\psi(x) = 0$ for ${|x|} \geq 2$. (For example, $\psi(x) = \frac{\phi(x + 2)} {\phi(x + 2) + \phi(-x - 1)} \frac{\phi(-x + 2)} {\phi(-x + 2) + \phi(x - 1)}$, where $\phi(x)$ is $\exp(-1/x)$ when $x \gt 0$ and otherwise vanishes.) Next, choose an infinite sequence $H = (H_n)_{n\geq{1}}$ of positive finite numbers:

\begin{displaymath}
H_n 
    = 
  \max_{0\leq{k}\lt{n}} 
  \,
  \max_{0\leq{i}\leq{k}} 
  \,
  \root{n-k}{
    2^{2n-i}
    \, 
    (k + 1)
    \, 
    n^{\underline{k}} 
    \,
    k^{\underline{i}} 
    \,
    i!^{-1} 
    \,
    {|c_n|} 
    \,
    {\|\psi^{(k-i)}\|_\infty}
   }
\end{displaymath}
(where $m^{\underline{j}}$ is the [[falling power]] $\prod_{0\leq{h}\lt{j}} (m - h)$, including the special case $m! = m^{\underline{m}}$); also, let $H_0 = 0$ (the [[bottom element]] of $[0,\infty]$, since $0 \leq k \lt 0$ never occurs). Finally, define

\begin{displaymath}
f(x) = \sum_{n=0}^\infty c_n \psi(H_n x) x^n .
\end{displaymath}
If things go well, the [[derivatives]] of $f$ will be

\begin{displaymath}
f^{(k)}(x) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(H_n x) x^{n-i} ,
\end{displaymath}
and I claim that this is so. Since $\psi^{(k-i)}(H_n x) = 0$ for ${|x|} \geq 2/H_n$, we have

\begin{displaymath}
\begin{aligned}
     {\|f^{(k)}\|_\infty} 
     & \leq 
    \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} H_n^{k-i} {\|\psi^{(k-i)}\|_\infty} (2/H_n)^{n-i} 
     \\
     & \leq \sum_{n=k}^\infty (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{n-i}/H_n^{n-k} 
     \\
     & \leq (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} k^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{k-i} + \sum_{n=k+1}^\infty 2^{-n} 
  \end{aligned}
  \,,
\end{displaymath}
which is finite. This proves [[uniform convergence]] of each claimed derivative (although not equiconvergence of all at once), and so each series not only converges but also may be antidifferentiated term by term, proving that $f^{(k)}$ is as claimed.

Finally (because $\psi^{(m)}(0) = 0$ for $m \gt 0$ but $\psi(0) = 1$, and the same goes for $0^m$),

\begin{displaymath}
f^{(k)}(0) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(0) 0^{n-i} = k!^{-1} k! k! c_k H_k^0 \psi(0) 0^0 = k! c_k ,
\end{displaymath}
making $c_k$ the $k$th coefficient of the Taylor series of $f$ at $0$, as desired.

\end{proof}
\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

\begin{itemize}%
\item the [[Hadamard lemma]]


\item the [[Tietze extension theorem]]


\item the [[Whitney extension theorem]]


\item the [[Steenrod-Wockel approximation theorem]]


\item [[embedding of smooth manifolds into formal duals of R-algebras]]


\item [[smooth Serre-Swan theorem]]


\item [[derivations of smooth functions are vector fields]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[asymptotic expansion]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original reference is

\begin{itemize}%
\item [[Émile Borel]], \emph{Sur quelques points de la th\'e{}orie des fonctions}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 3, 12 (1895), p. 9--55 \href{http://www.numdam.org/item?id=ASENS_1895_3_12__9_0}{numdam}

\end{itemize}
It had been actually proved by [[Guiseppe Peano]] before Borel

\begin{itemize}%
\item \'A{}d\'a{}m Besenyei, \emph{Peano's unnoticed proof of Borel's theorem} (\href{http://www.cs.elte.hu/~badam/publications/borel.pdf}{pdf})

\end{itemize}
Textbook discussion includes

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Gonzalo Reyes]], chapter I of \emph{[[Models for Smooth Infinitesimal Analysis]]}

\end{itemize}
A generalization to [[Banach spaces]] is in

\begin{itemize}%
\item John C. Wells, \emph{Differentiable functions on Banach spaces with Lipschitz derivatives}, J. Differential Geom. 8:1 (1973), 135-152 \href{http://projecteuclid.org/euclid.jdg/1214431488}{euclid}

\end{itemize}
and is cited (along with extensive discussion and (counter)examples) also as (Ch.III) 15.4 in

\begin{itemize}%
\item [[Andreas Kriegl]], [[Peter Michor]], \emph{[[The Convenient Setting of Global Analysis]]}, Mathematical Surveys and Monographs, 53 AMS (1997) \href{http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf}{pdf}

\end{itemize}
[[!redirects Borel's theorem]] [[!redirects Borel's theorem]] [[!redirects Borel/`s theorem]] [[!redirects Borel's theorem]] [[!redirects Borel's Theorem]] [[!redirects Borel's Theorem]] [[!redirects Borel/'s Theorem]] [[!redirects Borel's Theorem]]

[[!redirects Borel theorem]] [[!redirects Borel Theorem]]

[[!redirects Borel's theorem on power series]]

[[!redirects Borel's lemma]] [[!redirects Borel's lemma]] [[!redirects Borel/`s lemma]] [[!redirects Borel's lemma]] [[!redirects Borel's Lemma]] [[!redirects Borel's Lemma]] [[!redirects Borel/'s Lemma]] [[!redirects Borel's Lemma]]

[[!redirects Borel lemma]] [[!redirects Borel Lemma]]



\end{document}