  We introduce Banach spaces of vector-valued random variables motivated from
mathematical finance. So-called risk functionals are defined in a natural way
on these Banach spaces and it is shown that these functionals are Lipschitz
continuous. The risk functionals cannot be defined on strictly larger spaces of
random variables which creates a particular interest for the spaces presented.
We elaborate key properties of these Banach spaces and give representations of
their dual spaces in terms of vector measures with values in the dual space of
the state space.
