  We present the reduced basis method as a tool for developing emulators for
equations with tunable parameters within the context of the nuclear many-body
problem. The method uses a basis expansion informed by a set of solutions for a
few values of the model parameters and then projects the equations over a
well-chosen low-dimensional subspace. We connect some of the results in the
eigenvector continuation literature to the formalism of reduced basis methods
and show how these methods can be applied to a broad set of problems. As we
illustrate, the possible success of the formalism on such problems can be
diagnosed beforehand by a principal component analysis. We apply the reduced
basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic
trapping potential and to nuclear density functional theory for $^{48}$Ca,
achieving speed-ups of more than x150 in both cases when compared to
traditional solvers. The outstanding performance of the approach, together with
its straightforward implementation, show promise for its application to the
emulation of computationally demanding calculations, including uncertainty
quantification.
