  The first part of this paper deals with the topic of finding equivalent norms
and characterizations for vector-valued Besov and Triebel-Lizorkin spaces. We
will deduce general criteria by transferring and extending a theorem of Bui,
Paluszynski and Taibleson from the scalar to the vector-valued case. By using
special norms and characterizations we will derive necessary and sufficient
conditions for belonging to a vector-valued function spaces. It will be shown
that an element of the Schwartz space belongs to a function space if and only
if it can be written as a linear combination of harmonic atoms resp. quarks
with suitable conditions for the coefficients.
