nLab
Borel's theorem

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 $n+m\in \mathbb{N}$, write $C^{\mathrm{\infty }}({ℝ}^{n+m})$ for the $ℝ$-algebra of smooth functions with values in $ℝ$.

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

Theorem

Forming the Taylor series constitutes an isomorphism

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

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

Related theorems

• the Hadamard lemma

• the Tietze extension theorem

• the Steenrod-Wockel approximation theorem

References

Chapter I of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis