Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 13-19.

Let X be a complex Banach space. Recall that X admits a finite-dimensional Schauder decomposition if there exists a sequence {X n } n=1 ∞ of finite-dimensional subspaces of X, such that every x∈X has a unique representation of the form x=∑ n=1 ∞ x n , with x n ∈X n for every n. The finite-dimensional Schauder decomposition is said to be unconditional if, for every x∈X, the series x=∑ n=1 ∞ x n , which represents x, converges unconditionally, that is, ∑ n=1 ∞ x π(n) converges for every permutation π of the integers. For short, we say that X admits an unconditional F.D.D.We show that if X admits an unconditional F.D.D. then the following Runge approximation property holds:

Classification : 32H02 
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Meylan, Francine. Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 13-19. http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0/

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