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

Poincar\'e-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS