  In this paper, we revisit approximation properties of piecewise polynomial
spaces, which contain more than ${\cal P}_{r-1}$ but not ${\cal P}_r$. We
develop more accurate upper and lower error bounds that are sharper than those
used in literature. These new error bounds, especially the lower bounds are
particular useful to finite element methods. As an important application, we
establish sharp lower bounds of the discretization error for Laplace and
$2m$-th order elliptic eigenvalue problems in various finite element spaces
under shape regular triangulations, and investigate the asymptotic convergence
behavior for large numerical eigenvalue approximations.
