Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I ⊂ P ( N ) the quotient space l ∞ ( I ) / c 0 ( I ) is complete. The second main result states that consistently there is an admissible ideal I ⊂ P ( N ) such that the sets W ( I ) , of all real sequences with finite I -variation, and c ⁎ ( I ) , of all restrictively I -convergent sequences, are equal.