The Weierstrass approximation theorem for maps between real algebraic varieties

Abstract

P n(R), for some n, endowed with the Zariski topology and the sheaf of Rvalued regular functions. Morphisms between real algebraic varieties will be called regular maps. Every real algebraic variety carries also the Euclidean topology, that is, the topology induced by the usual metric topology on R. Unless explicitly stated otherwise, all topological notions related to real algebraic varieties will refer to the Euclidean topology. Given two real algebraic varieties X and Y , we regard the set R(X,Y ) of all regular maps from X into Y as a subset of the space C(X,Y ) of all continuous maps fromX into Y , endowed with the compact-open topology. If X and Y are nonsingular, then we also regard R(X,Y ) as a subset of the space C∞(X,Y ) of all C∞ maps from X into Y , endowed with the C∞ compact-open topology (the weak C∞ topology in the terminology used in [6]).