  Let G be a compact connected Lie group, and (M,\omega) a Hamiltonian G-space
with proper moment map \mu. We give a surjectivity result which expresses the
K-theory of the symplectic quotient M//G in terms of the equivariant K-theory
of the original manifold M, under certain technical conditions on \mu. This
result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in
symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott
lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian
G-spaces. We discuss this lemma in detail and highlight the differences between
the K-theory and rational cohomology versions of this lemma.
  We also introduce a K-theoretic version of equivariant formality and prove
that when the fundamental group of G is torsion-free, every compact Hamiltonian
G-space is equivariantly formal. Under these conditions, the forgetful map
K_{G}^{*}(M) \to K^{*}(M) is surjective, and thus every complex vector bundle
admits a stable equivariant structure. Furthermore, by considering complex line
bundles, we show that every integral cohomology class in H^{2}(M;\Z) admits an
equivariant extension in H_{G}^{2}(M;\Z).
