The structure of the Fr\'echet space $s$ regarding the series $\sum f_n\left(x_n\right)$

Abstract

We investigate the subsets of the Fr\'echet space $s$ of all sequences of real numbers equipped with the Fr\'echet metric $\rho$ from the Baire category
point of view. In particular, we concentrate on the "convergence" sets of the series $\sum f_n \left(x_n\right)$ that is, sets of sequences $x=(x_n)$
for which the series converges, or has a sum (perhaps infinite), or
oscillates. Provided all $f_n$ are continuous real functions, sufficient
conditions are given for the "convergence" sets to be of the first Baire
category or residual in $s$.