The class f of almost convergent sequences was introduced by G.G. Lorentz, using the idea of the Banach limits [A contribution to the theory of divergent sequences, Acta Math. 80(1948), 167-190]. Let f(0) ((B) over tilde) and f((B) over tilde) be the domain of the double sequential band matrix (B) over tilde((r) over tilde, (s) over tilde) in the sequence spaces f(0) and f. In this article, the beta- and gamma-duals of the space f(/3) are determined. Additionally, we give some inclusion theorems concerning with the spaces f(0)((B) over tilde) and f((B) over tilde). Moreover, the classes (f ((B) over tilde) :,mu,) and (mu : f((B) over tilde)) of infinite matrices are characterized, and the characterizations of some other classes are also given as an application of those main results, where mu is an arbitrary sequence space.