  We prove the injectivity and vanishing theorem for R-Hodge modules and
R-divisors over projective varieties, extending the results for rational Hodge
modules and integral divisors in \cite{Wu15}. In particular, the injectivity
generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing
Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for
Q-divisors. As a main application, we also deduce a Fujita-type freeness result
for R-Hodge modules in the normal crossing case.
