  We investigate tameness in the Fr\'echet spaces O(M) of analytic functions on
Stein manifolds M equipped with the compact open topology. Actually we will
look into tameness in the more general class of nuclear Fr\'echet spaces with
the properties weak DN and Omega, and then specialize to analytic function
spaces. We show that for a Stein manifold M, tameness of O(M) is equivalent to
the hyperconvexity of M.
