This is an announcement for the paper “Complemented basic sequences in Frechet spaces with finite dimensional decomposition” by Hasan Gülhttps://arxiv.org/find/math/1/au:+Gul_H/0/1/0/all/0/1, Süleyman Onalhttps://arxiv.org/find/math/1/au:+Onal_S/0/1/0/all/0/1.
Abstract: 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 $E$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.05049