# 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 ${ℝ}^{n+m}$ a Cartesian space of dimension $n+m\in ℕ$, write ${C}^{\infty }\left({ℝ}^{n+m}\right)$ for the $ℝ$-algebra of smooth functions with values in $ℝ$.

Write ${m}_{{ℝ}^{n}×\left\{0\right\}}^{\infty }\subset {C}^{\infty }\left({ℝ}^{n+m}\right)$ for the ideal of functions all whose partial derivatives along ${ℝ}^{m}$ vanish.

###### Theorem

Forming the Taylor series constitutes an isomorphism

${C}^{\infty }\left({ℝ}^{n+m}\right)/{m}_{{ℝ}^{n}×\left\{0\right\}}^{\infty }\stackrel{\simeq }{\to }{C}^{\infty }\left({ℝ}^{n}\right)\left[\left[{Y}_{1},\cdots ,{Y}_{m}\right]\right]$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 ${ℝ}^{m}$ vanish and the ring of power series in $m$-variables over ${C}^{\infty }\left({ℝ}^{n}\right)$.

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

## References

Chapter I of

Revised on October 12, 2012 13:11:53 by Ingo Blechschmidt (79.219.176.123)