  In this paper we introduce a Hilbert space-valued Malliavin calculus for
Poisson random measures. It is solely based on elementary principles from the
theory of point processes and basic moment estimates, and thus allows for a
simple treatment of the Malliavin operators. The main part of the theory is
developed for general Poisson random measures, defined on a $\sigma$-finite
measure space, with minimal conditions. The theory is shown to apply to a
space-time setting, suitable for studying stochastic partial differential
equations. As an application, we analyze the weak order of convergence of
space-time approximations for a class of linear equations with $\alpha$-stable
noise, $\alpha\in(1,2)$. For a suitable class of test functions, the weak order
of convergence is found to be $\alpha$ times the strong order.
