# nLab Borel's theorem

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

Borel’s theorem 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 actual partial derivatives.

## Statement

###### 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).

## 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 has been actually proved by Guiseppe Peano before Borel

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

Textbook discussion includes

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

Revised on April 2, 2015 16:15:27 by Urs Schreiber (195.113.30.252)