Summary: This paper considers the problem of optimal recovery of an element \(u\) of a Hilbert space \(\mathcal H\) from measurements of the form \(\ell_j(u)\), \(j=1,\dots,m\), where the \(\ell_j\) are known linear functionals on \(\mathcal H\). Problems of this type are well studied [C. A. Micchelli, T. J. Rivlin, and S. Winograd, Numer. Math., 26 (1976), pp. 191–200] and usually are carried out under an assumption that \(u\) belongs to a prescribed model class, typically a known compact subset of \(\mathcal H\). Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting, where the additional information about \(u\) is in the form of how well \(u\) can be approximated by a certain known subspace \(V_n\) of \(\mathcal H\) of dimension \(n\), or more generally, in the form of how well \(u\) can be approximated by each subspace from a sequence of nested subspaces \(V_0\subset V_1\cdots\subset V_n\) with each \(V_k\) of dimension \(k\). A recovery algorithm for the one-space formulation was proposed in [Y. Maday, A. T. Patera, J. D. Penn and M. Yano, Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933–965]. Their algorithm is proved, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem has a simple formulation if certain favorable bases are chosen to represent \(V_n\) and the measurements. The major contribution of the present paper is to analyze the multispace case by exploiting additional information derived from the whole hierarchy of spaces \(V_j\) rather than only from the largest space \(V_n\). It is shown that in this multispace case, the set of all \(u\) that satisfy the given information can be described as the intersection of a family of known ellipsoids in \(\mathcal H\). It follows that a near optimal recovery algorithm in the multispace problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of \(u\) in the multispace setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multispace problem, and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for \(u\).