  It is shown that a simple vertex operator algebra V is rational if and only
if its Zhu algebra A(V) is semisimple and each irreducible admissible V-module
is ordinary. A contravariant form on a Verma type admissible V-module is
constructed and the radical is exactly the maximal proper submodule. As an
application the rationality of V_L^+ for any positive definite even lattice is
obtained.
