  In the first part of this paper we discuss the completeness of two general
classes of weighted inductive limits of spaces of ultradifferentiable
functions. In the second part we study their duals and characterize these
spaces in terms of the growth of convolution averages of their elements. This
characterization gives a canonical way to define a locally convex topology on
these spaces and we give necessary and sufficient conditions for them to be
ultrabornological. In particular, our results apply to spaces of convolutors
for Gelfand-Shilov spaces.
