  We implement an infinite iteration scheme of Poincare-Dulac normal form
reductions to establish an energy estimate on the one-dimensional cubic
nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary
function space. This allows us to construct weak solutions of NLS in C_t
L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a
consequence of our construction, we also prove unconditional well-posedness of
NLS in H^s(T) for s \geq 1/6.
