  Discretization of the uniform norm of functions from a given finite
dimensional subspace of continuous functions is studied. Previous known results
show that for any $N$-dimensional subspace of the space of continuous functions
it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for
the uniform norm by the discrete norm and that one cannot improve on the
exponential growth of the number of sampling points for a good discretization
theorem in the uniform norm. In this paper we focus on two types of results,
which allow us to obtain good discretization of the uniform norm with
polynomial in $N$ number of points. In the first way we weaken the
discretization inequality by allowing a bound of the uniform norm by the
discrete norm multiplied by an extra factor, which may depend on $N$. In the
second way we impose restrictions on the finite dimensional subspace under
consideration. In particular, we prove a general result, which connects the
upper bound on the number of sampling points in the discretization theorem for
the uniform norm with the best $m$-term bilinear approximation of the Dirichlet
kernel associated with the given subspace.
