A quantitative version of a general subspace theorem is given. Let $K$ be an algebraic number field, $M(K)$ the set of absolute values $\|\,\|_v$ of $K$, let $S$ be a finite subset of $M(K)$, and for each $v\in S$, let $L_1^v,\ldots ,L_n^v$ be linearly independent linear forms in $n$ variables and with coefficients in $K$. Solutions $\beta \in K^n$ of the inequality $\prod _{v\in S}\prod _{i=1}^n\frac {\|L_i^v(\beta )\|_v}{\|L_i^v\|_v\|\beta \|_v} 0$, $H(\beta )$ denotes the height, and $\|\beta \|_v, \|L_i^v\|_v$ denote respectively the maximum norm of a vector $\beta $ and of the coefficient vector of a linear form. It is shown that these solutions are contained in the union of $t$ proper subspaces of $K^n$ and of a set “small” $\beta $s. Here $t$ is bounded in terms of $n, \deg K, \mathrm {card}\,S, \delta $ (but independently of the $L_i^v$), and the small $\beta $s have $H(\beta )$ under some effective bound (depending on the above data, as well as on the heights of the forms $L_i^v$). As applications, uniform estimates on the number of solutions of $S$-unit equations in $n$ variables are obtained