$\mathcal{F}$-bases with brackets and with individual brackets in Banach spaces

Abstract

We provide a partial answer to the question of Vladimir Kadets whether given an $\mathcal{F}$-basis of a Banach space $X$, with respect to some filter $\mathcal{F}\subset \mathcal{P}(\mathbb{N})$, the coordinate functionals are continuous. The answer is positive if the character of $\mathcal{F}$ is less than $\mathfrak{p}$. In this case every $\mathcal{F}$-basis with individual brackets is an $M$-basis with brackets determined by a set from $\mathcal{F}$.