  We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general
initial conditions in weighted Sobolev spaces that can support bright solitons
(but exclude spectral singularities corresponding to algebraic solitons). We
show that the set of such initial data is open and dense in a weighted Sobolev
space, and includes data of arbitrarily large $L^2$-norm. We prove global
well-posedness on this open and dense set. In a subsequent paper, we will use
these results and a steepest descent analysis to prove the soliton resolution
conjecture for the DNLS equation with the initial data considered here and
asymptotic stability of $N-$soliton solutions.
