  In this paper we study smooth structures on closed oriented 4-manifolds with
fundamental group Z_2 and definite intersection form. We construct infinitely
many irreducible, smooth, oriented, closed, definite four-manifolds with
fundamental group Z_2 and second Betti number 1 and 2. As an application, we
prove that when the second Betti number of a definite four-manifold with
fundamental group Z_2 is positive and it admits a smooth structure, then it
admits infinitely many smooth structures.
