Borel's theorem


Differential geometry

differential geometry

synthetic differential geometry








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 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.


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).


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 (