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 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:

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},
     pages = {13--19},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     mrnumber = {2240163},
     zbl = {1150.46017},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0/}
}
TY  - JOUR
AU  - Meylan, Francine
TI  - Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 13
EP  - 19
VL  - 5
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0/
LA  - en
ID  - ASNSP_2006_5_5_1_13_0
ER  - 
%0 Journal Article
%A Meylan, Francine
%T Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 13-19
%V 5
%N 1
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2006_5_5_1_13_0/
%G en
%F 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, Série 5, Tome 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 | Zbl

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

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

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

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

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

[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 | Zbl

[9] I. Patyi, On the ¯-equation in a Banach space, Bull. Soc. Math. France. 128 (2000), 391-406. | Numdam | MR | Zbl

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