In the present paper, the authors characterize (bv(p) : foo), (bv(p) : f), and (bv(p) : fo), where foo, f, and fo denote, respectively, the spaces of almost bounded, almost convergent and almost null sequences and the space bv(p) has recently been studied by Jarrah and Malkowsky (Collect. Math., 55:151-162, 2004).