  When a solution to the Cauchy problem for nonlinear dispersive equations is
obtained by a fixed point argument using auxiliary function spaces, it is
non-trivial to ensure uniqueness of solutions in a natural space such as the
class of continuous curves in the data space. This property is called
unconditional uniqueness, and proving it often requires some additional work.
In the last decade, unconditional uniqueness has been shown for some canonical
nonlinear dispersive equations by an integration-by-parts technique, which can
be regarded as a variant of the (Poincar\'e-Dulac) normal form reduction. In
this article, we aim to provide an abstract framework for establishing
unconditional uniqueness as well as existence of certain weak solutions via
infinite iteration of the normal form reduction. In particular, in an abstract
setting we find two sets of fundamental estimates, each of which can be used
repeatedly to generate all multilinear estimates of arbitrarily high degrees
required in this scheme. Then, we confirm versatility of the framework by
applying it to various equations, including the cubic nonlinear Schr\"odinger
equation (NLS) in higher dimension, the cubic NLS with fractional Laplacians,
the cubic derivative NLS, and the Zakharov system, for which new results on
unconditional uniqueness are obtained under the periodic boundary condition.
