The Lie group of real analytic diffeomorphisms is not real analytic

We construct an in finite-dimensional real analytic manifold structure on the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is de fined to be real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known, the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove that this group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.