Composition operators on spaces of real analytic functions

Let Ω 1 , Ω 2 be open subsets of ℝ   d   1and ℝ   d   2, respectively, and let A(Ω 1 ) denote the space of real analytic functions on Ω 1 . We prove a Glaeser type theorem by characterizing when a composition operator C φ : A(Ω 1 ) → A(Ω 2 ), C φ ( f ) ≔ f ∘ φ , is a topological embedding. Using this result we characterize when A(Ω 1 ) can be embedded topologically into A(Ω 2 ) as a locally convex space or as a topological algebra. We also characterize LB–subspaces and Fréchet subspaces of A(Ω 1 ). In particular, it follows that if A(Ω 1 ) and A(Ω 2 ) are isomorphic as locally convex spaces, then d 1 = d 2 .