  This paper is dedicated to the study of the derivative nonlinear
Schr\"odinger equation on the real line. The local well-posedness of this
equation in the Sobolev spaces is well understood since a couple of decades,
while the global well-posedness is not completely settled. For the latter
issue, the best known results up-to-date concern either Cauchy data in
$H^{\frac12}$ with mass strictly less than $4\pi$ or general initial conditions
in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the
derivative nonlinear Schr\"odinger equation is globally well-posed for general
Cauchy data in $H^{\frac12}$ and that furthermore the $H^{\frac12}$ norm of the
solutions remains globally bounded in time. One should recall that for $H^s$,
with $s < 1 / 2 $, the associated Cauchy problem is ill-posed in the sense that
uniform continuity with respect to the initial data fails. Thus, our result
closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is
achieved by combining the profile decomposition techniques with the
integrability structure of the equation.
