nLab
Borel's theorem

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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 C^\infty function at 0 n0\in\mathbb{R}^n to the ring of formal power series obtained by taking the Taylor series at 00 is surjective.

There are many extensions and variants.

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

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

Theorem

Forming the Taylor series constitutes an isomorphism

C ( n+m)/m n×{0} C ( n)[[Y 1,,Y m]] 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 m\mathbb{R}^m vanish and the ring of power series in mm-variables over C ( n)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)