Continuity of coordinate functionals for ideal Schauder basis

Given an ideal of subsets of natural numbers I we say that a sequence (x_n) is I-convergent to x if for every ε>0 condition {n \in N:d(x_n,x)>ε}\in I holds. We may use this notion to generalize the idea of Schauder basis, namely we say that a sequence (e_n) is an I-basis if for every x \in X there exists a unique sequence of scalars (\alpha_n) s.t. I sum of \alpha_n e_n equals x, which means that the sequence of partial sums is I-convergent to x. Once such a notion is introduced it is natural to ask whenever corresponding coordinate functionals are continuous. Such a question was posed by V. Kadets during the 4th conference Integration, Vector Measures, and Related Topics held in 2011 in Murcia. During my talk, I will discuss the problem and provide proof of continuity of considered functional. Those are joint results with Tomasz Kania and Noe de Rancourt, included in papers T. Kania, J. Swaczyna, Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces, Bulletin of the London Mathematical Society , 53 (1) (2021), 231-239, N. de Rancourt, T. Kania, J. Swaczyna, Continuity of coordinate functionals of filter bases in Banach spaces, Journal of Functional Analysis, 284 (9) (2023).