  We formulate a notion of K-stability for K\"ahler manifolds, and prove one
direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely,
we prove that the Mabuchi functional being bounded below (resp. coercive)
implies K-semistability (resp. uniformly K-stable). In particular this shows
that the existence of a constant scalar curvature K\"ahler metric implies
K-semistability, and K-stability if one assumes the automorphism group is
discrete. We also show how Stoppa's argument holds in the K\"ahler case, giving
a simpler proof of this K-stability statement.
