  We consider the Cauchy problem for the Gross-Pitaevskii infinite linear
hierarchy of equations on $\mathbb{R}^n.$ By introducing a (F)-norm in certain
Sobolev type spaces of sequences of marginal density matrices, we establish
local existence, uniqueness and stability of solutions. Explicit space-time
type estimates for the solutions are obtained as well. In particular, this
(F)-norm is compatible with the usual Sobolev space norm whenever the initial
data is factorized.
