  Weak lensing provides a direct measure of the distribution of mass in the
universe, and is therefore a uniquely powerful probe of dark matter. Weak
lensing can also be used to measure the twin phenomenon of dark energy, via its
effect upon the cosmological growth rate of structures. Essential for this
technique are well-resolved images of background galaxies out to large
distances. As a concrete example of the surveys that will become available by
the end of the decade, we consider the planned Supernova/Acceleration Probe
(SNAP) space telescope. Detailed simulations of space-based images,
manufactured using the shapelets formalism, enable us to quantitatively predict
the future sensitivity to weak lensing shear. The high number density of
galaxies resolved from space will enable maps of dark matter to be produced in
two and three dimensions, with a resolution superior to that from the ground.
Such observations will also afford reduced systematics for high-precision
measurements of weak lensing statistics. These will be used to set tight
constraints on cosmological parameters. In particular, the parameter for
equation of state of dark energy, w, will be measured using weak lensing with a
precision comparable to and somewhat orthogonal to constraints from other
methods.
