  Discretization of the uniform norm of functions from a given finite
dimensional subspace of continuous functions is studied. We pay special
attention to the case of trigonometric polynomials with frequencies from an
arbitrary finite set with fixed cardinality. We give two different proofs of
the fact 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. Previous known results show that one cannot improve
on the exponential growth of the number of sampling points for a good
discretization theorem in the uniform norm.
  Also, 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. We illustrate application of our technique on the example
of trigonometric polynomials.
