  Assume that $V_h$ is a space of piecewise polynomials of degree less than
$r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the
following well-known upper bound holds for a sufficiently smooth function $u$
and $p\in [1, \infty]$ $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,\Omega,h} \le C
h^{r-j} |u|_{r,p,\Omega},\quad 0\le j\le r. $$ In this paper, we prove that,
roughly speaking, if $u\not\in V_h$, the above estimate is sharp. Namely, $$
  \inf_{v_h\in V_h}\|u-v_h\|_{j,p,\Omega,h} \ge c h^{r-j},\quad 0\le j\le r, \
\ 1\leq p\leq \infty, $$ for some $c>0$.
  The above result is further extended to various situations including more
general Sobolev space norms, general shape regular grids and many different
types of finite element spaces. As an application, the sharpness of finite
element approximation of elliptic problems and the corresponding eigenvalue
problems is established.
