On the Borel Theorem
by
J. Schmets
Université de Liège
Coauthors: M. Valdivia (University of Valencia)

The original Borel theorem reads as follows:

For every sequence r₀, r₁, ... of real numbers, there is a real 2 \pi-periodic and C_\infty-function f on the real line such that f^(m)(0) = r_m for every m.

An enhanced version of this result states that one may require f to be a real-analytic function outside the origin, up to the loss of the 2 \pi-periodicity.

The extension of the original result to the setting of locally convex spaces has been considered by different authors.

The enhanced version can be generalized as follows:

Let X be a real Banach space. If A₀ is a real number and if, for every integer m >= 1, A_m is a m-linear symmetric approximable and real functional on X^m, then there is a real C_\infty-function f on X with the following properties:

1. f^(j)(0) = A_j for every j in {0, 1, 2, ... },
2. f^(j)(x) is approximable for every positive integer j and x in X,
3. f^(j) is bounded on every bounded subset of X for every integer j >= 0,
4. f is real analytic on X \{0} endowed with the topology of the uniform convergence on the compact subsets of X^*.

Date received: December 30, 1995

Copyright © 1995 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caad-02.