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.
For a Cartesian space of dimension , write for the -algebra of smooth functions with values in .
Write for the ideal of functions all whose partial derivatives along vanish.
Forming the Taylor series constitutes an isomorphism
between smooth functions modulo those whose derivatives along vanish and the ring of power series in -variables over .
This appears for instance as (Moerdijk-Reyes, theorem I.1.3).
Chapter I of
Revised on March 10, 2015 13:50:55
by Urs Schreiber