Let be a complex Banach space. Recall that admits a finite-dimensional Schauder decomposition if there exists a sequence of finite-dimensional subspaces of such that every has a unique representation of the form with for every The finite-dimensional Schauder decomposition is said to be unconditional if, for every the series which represents converges unconditionally, that is, converges for every permutation of the integers. For short, we say that 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}, 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] “Complex Analysis on Infinite Dimensional Spaces”, Springer, Berlin, 1999. | MR | Zbl
,[2] “Linear Operators” I, John Wiley and Sons, New York, 1988. | Zbl
and ,[3] Approximation of holomorphic functions in certain Banach spaces, Internat. J. Math. 15 (2004), 467-471. | Zbl
,[4] Approximation de fonctions holomorphes d'un nombre infini de variables, Ann. Inst. Fourier (Grenoble) 49 (1999), 1293-1304. | Numdam | MR | Zbl
,[5] The Dolbeaut complex in infinite dimensions, III, Invent. Math. 142 (2000), 579-603. | MR | Zbl
,[6] Approximation of holomorphic functions of infinitely many variables, Ann. Inst. Fourier (Grenoble) 50 (2000), 423-442. | Numdam | MR | Zbl
,[7] Seminar given at Purdue University, 2004.
,[8] “Classical Banach Spaces I, Sequence Spaces”, Springer-Verlag, Berlin Heidelberg New York., Vol. 92, 1977. | MR | Zbl
and ,[9] On the -equation in a Banach space, Bull. Soc. Math. France. 128 (2000), 391-406. | Numdam | MR | Zbl
,[10] “Bases in Banach Spaces”, I-II, Springer, Berlin, 1981. | Zbl
,