nLab Borel's theorem

Idea

Borel's Theorem (also called 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.

Statements

Theorem

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.

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 $\mathbb{R}$-algebra of smooth functions 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.

Theorem

Forming the Taylor series constitutes an isomorphism

$$C^\infty(\mathbb{R}^{n+m})/m^\infty_{\mathbb{R}^n \times \{0\}} \stackrel{\simeq}{\longrightarrow} C^\infty(\mathbb{R}^n) [ [ Y_1, \cdots, Y_m] ]$$

between smooth functions modulo those whose derivatives along $\mathbb{R}^m$ vanish and the ring of power series in $m$-variables over $C^\infty(\mathbb{R}^n)$.

This appears for instance as (Moerdijk-Reyes, theorem I.1.3).

Proof

There is a full proof in Moerdijk–Reyes, cited above. Here we prove Theorem to indicate the method; the general version is not substantially different. (This is based on the proof in the English Wikipedia article at the time of writing, but with more details.)

Proof (of Theorem )

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

$$f(x) = \sum_{n=0}^\infty c_n x^n ,$$

but this is only correct if the sum 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 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:

$$H_n = \max_{0\leq{k}\lt{n}} \, \max_{0\leq{i}\leq{k}} \, \root{n-k}{ 2^{2n-i} \, (k + 1) \, n^{\underline{k}} \, k^{\underline{i}} \, i!^{-1} \, {|c_n|} \, {\|\psi^{(k-i)}\|_\infty} }$$

(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

$$f(x) = \sum_{n=0}^\infty c_n \psi(H_n x) x^n .$$

If things go well, the derivatives of $f$ will be

$$f^{(k)}(x) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(H_n x) x^{n-i} ,$$

and I claim that this is so. Since $\psi^{(k-i)}(H_n x) = 0$ for ${|x|} \geq 2/H_n$, we have

$$\begin{aligned} {\|f^{(k)}\|_\infty} & \leq \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} H_n^{k-i} {\|\psi^{(k-i)}\|_\infty} (2/H_n)^{n-i} \\ & \leq \sum_{n=k}^\infty (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{n-i}/H_n^{n-k} \\ & \leq (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} k^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{k-i} + \sum_{n=k+1}^\infty 2^{-n} \end{aligned} \,,$$

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 $0^m$),

$$f^{(k)}(0) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(0) 0^{n-i} = k!^{-1} k! k! c_k H_k^0 \psi(0) 0^0 = k! c_k ,$$

making $c_k$ the $k$th coefficient of the Taylor series of $f$ at $0$, as desired.

Related theorems

• the Hadamard lemma

• the Tietze extension theorem

• the Whitney extension theorem

• the Steenrod-Wockel approximation theorem

• embedding of smooth manifolds into formal duals of R-algebras

• smooth Serre-Swan theorem

• derivations of smooth functions are vector fields

Related concepts

• asymptotic expansion

References

The original reference is

• Émile Borel, Sur quelques points de la théorie des fonctions, Annales scientifiques de l’École Normale Supérieure, Sér. 3, 12 (1895),

  p. 9–55 numdam

It had been actually proved by Giuseppe Peano before Borel

• Ádám Besenyei, Peano’s unnoticed proof of Borel’s theorem (pdf)

Textbook discussion includes

• Ieke Moerdijk, Gonzalo Reyes, chapter I of Models for Smooth Infinitesimal Analysis

A generalization to Banach spaces is in

• John C. Wells, Differentiable functions on Banach spaces with Lipschitz derivatives, J. Differential Geom. 8:1 (1973), 135-152 euclid

and is cited (along with extensive discussion and (counter)examples) also as (Ch.III) 15.4 in

• Andreas Kriegl, Peter Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53 AMS (1997) pdf