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

TL;DR

This paper investigates conditions under which $$-bases in Banach spaces have continuous coordinate functionals, showing positivity when the filter's character is below a certain cardinality, linking $$-bases to $M$-bases.

Contribution

It provides a partial answer to Kadets' question by establishing continuity of coordinate functionals for $$-bases with specific filter character constraints.

Findings

Coordinate functionals are continuous if the filter's character is less than $rak{p}$.

Every $$-basis with individual brackets is an $M$-basis with brackets from the filter.

The result connects the structure of $$-bases to classical $M$-bases under certain conditions.

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 $\mathcal F$.