Abstract

Let $X$ be a complex Banach space. Recall that $X$ admits a finite-dimensional Schauder decomposition if 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 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 }$ 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