Borel's Theorem (also called Borel's Lemma) 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 its actual derivatives.
The canonical map from the ring of germs of function at to the ring of formal power series obtained by taking the Taylor series at is surjective.
There are many extensions and variants.
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).
There is a full proof in Moerdijk–Reyes, cited above. Here we prove Theorem 1 to indicate the method; the general version is not substantially different. (This is based on the proof in the English Wikipedia article at the time of writing, but with more details.)
Proof (of Theorem 1)
A real power series at is given simply by an infinite sequence of real numbers. Given such a sequence, we would ideally use
but this is only correct if the sum converges on at least some neighbourhood of (in other words if the power series has a positive radius of convergence).
To ensure this, let be a smooth bump function chosen so that for and for . (For example, , where is when and otherwise vanishes.) Next, choose an infinite sequence of positive finite numbers:
(where is the falling power , including the special case ); also, let (the bottom element of , since never occurs). Finally, define
If things go well, the derivatives of will be
and I claim that this is so. Since for , we have
which is finite. This proves uniform convergence of each claimed derivative (although not equiconvergence of all at once), and so each series not only converges but also may be antidifferentiated term by term, proving that is as claimed.
Finally (because for but , and the same goes for ),
making the th coefficient of the Taylor series of at , as desired.
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 had 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