  In this article we introduce a new class of weighted sequence spaces of
Sobolev type and prove several compact embedding theorems for them. In all
cases our proofs are based on the existence of a Schauder basis that spans all
of the spaces under consideration. Our choice of spaces is primarily motivated
by earlier work on infinite-dimensional dynamical systems of hyperbolic type
arising in non-equilibrium statistical mechanics. We also prove a theorem of
Pitt's type asserting that under some conditions every linear bounded
transformation from one weighted sequence space into another is compact.
