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)