# Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition

• Volume: 5, Issue: 1, page 13-19
• ISSN: 0391-173X

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## Abstract

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Let $X$ be a complex Banach space. Recall that $X$ admits afinite-dimensional Schauder decompositionif 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 beunconditionalif, 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:

## How to cite

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Meylan, Francine. "Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.1 (2006): 13-19. <http://eudml.org/doc/239731>.

@article{Meylan2006,
abstract = {Let $X$ be a complex Banach space. Recall that $X$ admits afinite-dimensional Schauder decompositionif there exists a sequence $\{\lbrace X_n\rbrace \}_\{n=1\}^\{\infty \}$ 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 beunconditionalif, for every $x \in X,$ the series $x= \sum _\{n=1\}^\{\infty \}x_n,$ which represents $x,$ converges unconditionally, that is, $\sum _\{n=1\}^\{\infty \}x_\{\pi (n)\}$ 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:},
author = {Meylan, Francine},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Runge type approximation; complex analysis on Banach spaces},
language = {eng},
number = {1},
pages = {13-19},
publisher = {Scuola Normale Superiore, Pisa},
title = {Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition},
url = {http://eudml.org/doc/239731},
volume = {5},
year = {2006},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 1
SP - 13
EP - 19
AB - Let $X$ be a complex Banach space. Recall that $X$ admits afinite-dimensional Schauder decompositionif there exists a sequence ${\lbrace X_n\rbrace }_{n=1}^{\infty }$ 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 beunconditionalif, for every $x \in X,$ the series $x= \sum _{n=1}^{\infty }x_n,$ which represents $x,$ converges unconditionally, that is, $\sum _{n=1}^{\infty }x_{\pi (n)}$ 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:
LA - eng
KW - Runge type approximation; complex analysis on Banach spaces
UR - http://eudml.org/doc/239731
ER -

## References

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7. [7] L. Lempert, Seminar given at Purdue University, 2004.
8. [8] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I, Sequence Spaces”, Springer-Verlag, Berlin Heidelberg New York., Vol. 92, 1977. Zbl0362.46013MR500056
9. [9] I. Patyi, On the $\overline{\partial }$-equation in a Banach space, Bull. Soc. Math. France. 128 (2000), 391–406. Zbl0967.32036MR1792475
10. [10] I. Singer, “Bases in Banach Spaces”, I-II, Springer, Berlin, 1981. Zbl0198.16601

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