  We approximate functions defined on smooth bounded domains by elements of the
eigenspaces of the Laplacian or the Stokes operator in such a way that the
approximations are bounded and converge in both Sobolev and Lebesgue spaces. We
prove an abstract result referred to fractional power spaces of positive,
self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our
main result by using the explicit form of these fractional power spaces for the
Dirichlet Laplacian and Stokes operators. As a simple application, we prove
that all weak solutions of the incompressible convective Brinkman--Forchheimer
equations posed on a bounded domain in ${\mathbb R}^3$ satisfy the energy
equality.
