  The well known Weyl's Law (Weyl's asymptotic formula) gives an approximation
to the number $\mathcal{N}_{\omega}$ of eigenvalues (counted with
multiplicities) on a large interval $[0,\>\omega]$ of the Laplace-Beltrami
operator on a compact Riemannian manifold ${\bf M}$. In this paper we prove a
kind of a weak version of the Weyl's law on certain compact metric measure
spaces ${\bf X}$ which are equipped with a self-adjoint non-negative operator
$\mathcal{L}$ acting in $L_{2}({\bf X})$.
  Roughly speaking, we show that if a certain Poincar\'e inequality holds then
$\mathcal{N}_{\omega}$ is controlled by the cardinality of an appropriate cover
$\mathcal{B}_{\omega^{-1/2}}=\{B(x_{j},\omega^{-1/2})\},\>\>\>x_{j}\in {\bf
X},$ of ${\bf X}$ by balls of radius $\omega^{-1/2}$.
  Moreover, an opposite inequality holds if the heat kernel that corresponds to
$\mathcal{L}$ satisfies short time Gaussian estimates.
  It is known that in the case of the so-called strongly local regular with a
complete intrinsic metric Dirichlet spaces the Poincar\'e inequality holds iff
the corresponding heat kernel satisfies short time Gaussian estimates. Thus for
such spaces one obtains that $\mathcal{N}_{\omega}$ is essentially equivalent
to the cardinality of a cover $\mathcal{B}_{\omega^{-1/2}}$.
