  We study weighted $(PLB)$-spaces of ultradifferentiable functions defined via
a weight function (in the sense of Braun, Meise and Taylor) and a weight
system. We characterize when such spaces are ultrabornological in terms of the
defining weight system. This generalizes Grothendieck's classical result that
the space $\mathcal{O}_M$ of slowly increasing smooth functions is
ultrabornological to the context of ultradifferentiable functions. Furthermore,
we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the
above result, characterize when such spaces are ultrabornological. In
particular, we show that the multiplier space of the space of Fourier
ultrahyperfunctions is ultrabornological, whereas the one of the space of
Fourier hyperfunctions is not.
