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

A dimension-free estimate on $L^2$ for the maximal Riesz transform in terms of the Riesz transform