  We study approximation classes for adaptive time-stepping finite element
methods for time-dependent Partial Differential Equations (PDE). We measure the
approximation error in $L_2([0,T)\times\Omega)$ and consider the approximation
with discontinuous finite elements in time and continuous finite elements in
space, of any degree. As a byproduct we define Besov spaces for vector-valued
functions on an interval and derive some embeddings, as well as Jackson- and
Whitney-type estimates.
