Remarks on ideal boundedness, convergence and variation of sequences (Q615907)

The authors answer a questions asked by \textit{A. Faisant} et al. in [J. Math. Anal. Appl. 306, No. 2, 432--439 (2005; Zbl 1070.40001)]. Let \(P(\mathbb N)\) be the power set of \(\mathbb N\). The first main result states that for every admissible ideal \({\mathcal I} \subset P(\mathbb N)\) the quotient space \(l^\infty({\mathcal I})/c_0({\mathcal I})\), where \(l^\infty({\mathcal I})\) is the vector space of all \({\mathcal I}\)-bounded sequences \(x\in\mathbb R^{\mathbb N}\) and \(c_0({\mathcal I})\) is the set of all sequences \({\mathcal I}\)-convergent to 0, is complete. The second main result expresses that consistently there is an admissible ideal \({\mathcal I} \subset P(\mathbb N)\) such that the set \(W({\mathcal I})\) of all real sequences with finite \({\mathcal I}\)-variation and the set \(c^*({\mathcal I})\) of all restrictively \({\mathcal I}\)-convergent sequences are equal.