# 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

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

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

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

Chapter I of

Revised on September 11, 2013 10:46:19 by Urs Schreiber (82.169.114.243)