Complemented basic sequences in Frechet spaces with finite dimensional decomposition

TL;DR

This paper proves that in certain Frechet-Montel spaces with finite dimensional unconditional decompositions, sequences formed by choosing one element from each subspace have subsequences that are complemented in the entire space.

Contribution

It establishes the existence of complemented subsequences in Frechet-Montel spaces with finite dimensional unconditional decompositions, extending previous results to this class of spaces.

Findings

Sequences have complemented subsequences in the space.

Finite dimensional unconditional decompositions facilitate the construction.

Results apply to Frechet-Montel spaces with bounded dimension subspaces.

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 $E$