Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, p. 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 xX 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 xX, 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:

@article{ASNSP_2006_5_5_1_13_0,
     author = {Meylan, Francine},
     title = {Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     pages = {13-19},
     zbl = {1150.46017},
     mrnumber = {2240163},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0}
}
Meylan, Francine. Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, pp. 13-19. http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0/

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