  W prove a dimension-free estimate for the $L^2(\mathbb{R}^d)$ norm of the
maximal truncated Riesz transform in terms of the $L^2(\mathbb{R}^d)$ norm of
the Riesz transform. Consequently, the vector of maximal truncated Riesz
transforms has a dimension-free estimate on $L^2(\mathbb{R}^d)$. We also show
that the maximal function of the vector of truncated Riesz transforms has a
dimension-free estimate on all $L^p(\mathbb{R}^d)$ spaces, $1<p<\infty.$
