Approximation theorems for Nash mappings and Nash manifolds

Abstract

Let 0 ≤ r > ∞ 0 \leq r > \infty . A Nash function on R n {\mathbf {R}^n} is a C r {C^r} function whose graph is semialgebraic. It is shown that a C r {C^r} Nash function is approximated by a C ω {C^\omega } Nash one in a strong topology defined in the same way as the usual topology on the space S \mathcal {S} of rapidly decreasing C ∞ {C^\infty } functions. A C r {C^r} Nash manifold in R n {\mathbf {R}^n} is a semialgebraic C r {C^r} manifold. We also prove that a C r {C^r} Nash manifold for r ≥ 1 r \ge 1 is approximated by a C ω {C^\omega } Nash manifold, from which we can classify all C r {C^r} Nash manifolds by C r {C^r} Nash diffeomorphisms.