  We develop the existence, uniqueness, continuity, stability, and scattering
theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n
\geq 3$, for solutions which have large, but finite, energy and large, but
finite, Strichartz norms. For dimensions $n \leq 6$, this theory is a standard
extension of the small data well-posedness theory based on iteration in
Strichartz spaces. However, in dimensions $n > 6$ there is an obstruction to
this approach because of the subquadratic nature of the nonlinearity (which
makes the derivative of the nonlinearity non-Lipschitz). We resolve this by
iterating in exotic Strichartz spaces instead. The theory developed here will
be applied in a subsequent paper of the second author, to establish global
well-posedness and scattering for the defocusing energy-critical equation for
large energy data.
