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 ${\left\{{X}_{n}\right\}}_{n=1}^{\infty }$ of finite-dimensional subspaces of $X,$ such that every $x\in X$ has a unique representation of the form $x={\sum }_{n=1}^{\infty }{x}_{n},$ with ${x}_{n}\in {X}_{n}$ for every $n.$ The finite-dimensional Schauder decomposition is said to be unconditional if, for every $x\in X,$ the series $x={\sum }_{n=1}^{\infty }{x}_{n},$ which represents $x,$ converges unconditionally, that is, ${\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}$ converges for every permutation $\pi$ 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
@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/

[1] S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer, Berlin, 1999. | MR 1705327 | Zbl 1034.46504

[2] N. Dunford and T. Schwartz, “Linear Operators” I, John Wiley and Sons, New York, 1988. | Zbl 0084.10402

[3] B. Josefson, Approximation of holomorphic functions in certain Banach spaces, Internat. J. Math. 15 (2004), 467-471. | Zbl 1061.46041

[4] L. Lempert, Approximation de fonctions holomorphes d'un nombre infini de variables, Ann. Inst. Fourier (Grenoble) 49 (1999), 1293-1304. | Numdam | MR 1703089 | Zbl 0944.46046

[5] L. Lempert, The Dolbeaut complex in infinite dimensions, III, Invent. Math. 142 (2000), 579-603. | MR 1804162 | Zbl 0983.32010

[6] L. Lempert, Approximation of holomorphic functions of infinitely many variables, Ann. Inst. Fourier (Grenoble) 50 (2000), 423-442. | Numdam | MR 1775356 | Zbl 0969.46032

[7] L. Lempert, Seminar given at Purdue University, 2004.

[8] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I, Sequence Spaces”, Springer-Verlag, Berlin Heidelberg New York., Vol. 92, 1977. | MR 500056 | Zbl 0362.46013

[9] I. Patyi, On the $\overline{\partial }$-equation in a Banach space, Bull. Soc. Math. France. 128 (2000), 391-406. | Numdam | MR 1792475 | Zbl 0967.32036

[10] I. Singer, “Bases in Banach Spaces”, I-II, Springer, Berlin, 1981. | Zbl 0198.16601