  For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform
operators on metric measure spaces with Riemannian Ricci curvature bounded from
below, without any restriction on their dimension. This large class of spaces
include e.g. that of Hilbert spaces endowed with a log-concave probability
measure. As a consequence, we extend the range of the validity of the
Lusin-type approximation of Sobolev by Lipschitz functions, previously obtained
by L. Ambrosio, E. Bru\`e and the third author in the quadratic case, i.e.
$p=2$. The proofs are analytic and rely on computations on an explicit Bellman
function.
