  Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$. In this paper
we introduce sufficient conditions for the convergence uniform on compacts in
probability for a sequence of $\Phi'$-valued processes with continuous or
c\`{a}dl\`{a}g paths. We illustrate the usefulness of our results by
considering two applications to stochastic analysis. First, we introduce a
topology on the space of $\Phi'$-valued semimartingales which are good
integrators and show that this topology is complete and that the stochastic
integral mapping is continuous on the integrators. Second, we introduce
sufficient conditions for the convergence uniform on compacts in probability of
the solutions to a sequence of linear stochastic evolution equations driven by
semimartingale noise.
