# Borel's lemma

In mathematics, **Borel's lemma**, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.

## Contents

## Statement

Suppose *U* is an open set in the Euclidean space **R**^{n}, and suppose that *f*_{0}, *f*_{1} ... is a sequence of smooth functions on *U*.

If *I* is an any open interval in **R** containing 0 (possibly *I* = **R**), then there exists a smooth function *F*(*t*, *x*) defined on *I*×*U*, such that

for *k* ≥ 0 and *x* in *U*.

## Proof

Proofs of Borel's lemma can be found in many text books on analysis, including Template:Harvtxt and Template:Harvtxt, from which the proof below is taken.

Note that it suffices to prove the result for a small interval *I* = (−ε,ε), since if ψ(*t*) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(*t*) ⋅ *F*(*t*, *x*) gives a solution on **R** × *U*. Similarly using a smooth partition of unity on **R**^{n} subordinate to a covering by open balls with centres at δ⋅**Z**^{n}, it can be assumed that all the *f*_{m} have compact support in some fixed closed ball *C*. For each *m*, let

where ε_{m} is chosen sufficiently small that

for |α| < *m*. These estimates imply that each sum

is uniformly convergent and hence that

is a smooth function with

By construction

**Note:** Exactly the same construction can be applied, without the auxiliary space *U*, to produce a smooth function on the interval *I* for which the derivatives at 0 form an arbitrary sequence.

## See also

## References

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*This article incorporates material from Borel lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*