Borel's theorem

*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.

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.

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).

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

- 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

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