ABSTRACT. A theory of best approximation with interpolatory contraints from a finite-dimensional subspace M of a normed linear space X is developed. In particular, to each x ∈ X, best approximations are sought from a subset M(x) of M which depends on the element x being approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certain fixed subspace M_{0} of M. More detailed results can be obtained when (1) X is a Hilbert space, or (2) M is an “interpolating subspace” of X (in the sense of [1]).