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...