  In this article, Fefferman-Stein inequalities in $L^p(\mathbb R^d;\ell^q)$
withbounds independent of the dimension $d$ are proved, for all $1 \textless{}
p, q \textless{} + \infty.$This result generalizes in a vector-valued setting
the famous one by Steinfor the standard Hardy-Littlewood maximal operator. We
then extendour result by replacing $\ell^q$ with an arbitrary UMD Banach
lattice. Finally,we prove similar dimensionless inequalities in the setting of
the Grushinoperators.
