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Let $X$ be a compact normal complex space of dimension $n$ and $L$ be aholomorphic line bundle on $X$. Suppose that$\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinctirreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ isan $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the spaceof holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least$\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is anirreducible analytic subset of dimension $m$, we consider the space $H^0_0(X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to globalholomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet$(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we givea general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When$L$ is endowed with a continuous Hermitian metric, we show that theFubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certainequilibrium current on $Y$. We apply this to the study of the equidistributionof zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as$p\to\infty$.

Restricted spaces of holomorphic sections vanishing along subvarieties