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Borel's theorem

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Differential geometry

differential geometry

synthetic differential geometry

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Contents

Idea

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.

Statements

Theorem

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.

Theorem

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).

Proof

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 00 is given simply by an infinite sequence c=(c n) n0c = (c_n)_{n\geq{0}} of real numbers. Given such a sequence, we would ideally use

f(x)= n=0 c nx n, f(x) = \sum_{n=0}^\infty c_n x^n ,

but this is only correct if the sum converges on at least some neighbourhood of 00 (in other words if the power series has a positive radius of convergence).

To ensure this, let ψ\psi be a smooth bump function chosen so that ψ(x)=1\psi(x) = 1 for |x|1{|x|} \leq 1 and ψ(x)=0\psi(x) = 0 for |x|2{|x|} \geq 2. (For example, ψ(x)=ϕ(x+2)ϕ(x+2)+ϕ(x1)ϕ(x+2)ϕ(x+2)+ϕ(x1)\psi(x) = \frac{\phi(x + 2)} {\phi(x + 2) + \phi(-x - 1)} \frac{\phi(-x + 2)} {\phi(-x + 2) + \phi(x - 1)}, where ϕ(x)\phi(x) is exp(1/x)\exp(-1/x) when x>0x \gt 0 and otherwise vanishes.) Next, choose an infinite sequence H=(H n) n1H = (H_n)_{n\geq{1}} of positive finite numbers:

H n=max 0k<nmax 0ik2 2ni(k+1)n k̲k i̲i! 1|c n|ψ (ki) nk H_n = \max_{0\leq{k}\lt{n}} \, \max_{0\leq{i}\leq{k}} \, \root{n-k}{ 2^{2n-i} \, (k + 1) \, n^{\underline{k}} \, k^{\underline{i}} \, i!^{-1} \, {|c_n|} \, {\|\psi^{(k-i)}\|_\infty} }

(where m j̲m^{\underline{j}} is the falling power 0h<j(mh)\prod_{0\leq{h}\lt{j}} (m - h), including the special case m!=m m̲m! = m^{\underline{m}}); also, let H 0=0H_0 = 0 (the bottom element of [0,][0,\infty], since 0k<00 \leq k \lt 0 never occurs). Finally, define

f(x)= n=0 c nψ(H nx)x n. f(x) = \sum_{n=0}^\infty c_n \psi(H_n x) x^n .

If things go well, the derivatives of ff will be

f (k)(x)= n=k i=0 ki! 1k i̲n i̲c nH n kiψ (ki)(H nx)x ni, f^{(k)}(x) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(H_n x) x^{n-i} ,

and I claim that this is so. Since ψ (ki)(H nx)=0\psi^{(k-i)}(H_n x) = 0 for |x|2/H n{|x|} \geq 2/H_n, we have

f (k) n=k i=0 ki! 1k i̲n i̲|c n|H n kiψ (ki) (2/H n) ni n=k (k+1)max 0iki! 1k i̲n i̲|c n|ψ (ki) 2 ni/H n nk (k+1)max 0iki! 1k i̲k i̲|c n|ψ (ki) 2 ki+ n=k+1 2 n, \begin{aligned} {\|f^{(k)}\|_\infty} & \leq \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} H_n^{k-i} {\|\psi^{(k-i)}\|_\infty} (2/H_n)^{n-i} \\ & \leq \sum_{n=k}^\infty (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{n-i}/H_n^{n-k} \\ & \leq (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} k^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{k-i} + \sum_{n=k+1}^\infty 2^{-n} \end{aligned} \,,

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 f (k)f^{(k)} is as claimed.

Finally (because ψ (m)(0)=0\psi^{(m)}(0) = 0 for m>0m \gt 0 but ψ(0)=1\psi(0) = 1, and the same goes for 0 m0^m),

f (k)(0)= n=k i=0 ki! 1k i̲n i̲c nH n kiψ (ki)(0)0 ni=k! 1k!k!c kH k 0ψ(0)0 0=k!c k, f^{(k)}(0) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(0) 0^{n-i} = k!^{-1} k! k! c_k H_k^0 \psi(0) 0^0 = k! c_k ,

making c kc_k the kkth coefficient of the Taylor series of ff at 00, as desired.

References

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

Revised on February 15, 2017 16:18:10 by Toby Bartels (64.89.54.7)