Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition

Meylan, Francine

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 ${\left\{X_n\right\}}_{n=1}^{\infty }$ 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

[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 $\overline{\partial }$-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