Bochner's inequality is one of the most fundamental estimates in geometric analysis on Riemannian manifolds. It states that
for each smooth function u on a Riemannian manifold provided K is a lower bound for the Ricci curvature on and N is an upper bound for the dimension. The main result we present in this talk is a Bochner inequality on metric measure spaces with linear heat flow and satisfying the (reduced) curvature-dimension condition. Indeed, we will also prove the converse: if the heat flow on a metric measure space is linear then an appropriate version of the Bochner inequality (for the canonical gradient and Laplacian) will imply the reduced curvature-dimension condition. Besides that, we also derive new, sharp Wasserstein-contraction results for the heat flow as well as Bakry-Ledoux type gradient estimates and prove that each of them is equivalent to the curvature-dimension condition.