  In this paper we consider stochastic integration with respect to cylindrical
Brownian motion in infinite dimensional spaces. We study weak characterizations
of stochastic integrability and present a natural continuation of results of
van Neerven, Weis and the second named author. The limitation of weak
characterizations will be demonstrated with a nontrivial counterexample. The
second subject treated in the paper addresses representation theory for random
variables in terms of stochastic integrals. In particular, we provide an
infinite dimensional version of Dudley's representation theorem for random
variables and an extension of Doob's representation for martingales.
