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

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

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

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topMeylan, 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 -

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- [8] J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I, Sequence Spaces”, Springer-Verlag, Berlin Heidelberg New York., Vol. 92, 1977. Zbl0362.46013MR500056
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