Complemented basic sequences in Frechet spaces with finite dimensional   decomposition
  Let $E$ be a Frechet-Montel space and $(E_n)_{n \in \mathbb{N}}$ be a finite dimensional unconditional decomposition of $E$ with $\dim(E_n)\leq k$ for some fixed $k \in \mathbb{N}$ and for all $n \in \mathbb{N}$. Consider a sequence $(x_n)_{n \in \mathbb{N}}$ formed by taking an element $x_n$ from each $E_n$ for all $n \in \mathbb{N}$. Then $(x_n)_{n \in \mathbb{N}}$ has a subsequence which is complemented in $E$ 
Hasan G\"ul and S\"uleyman Onal
math.FA