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(1) for every xE U, there exis ts fEA such t h a t f ( x ) r (2) for every pair x, yEU, with xr there exists fEA such that f(x)r (3) for every xE U and vEE, with v ~ 0 , there exists fEA such that Df(x)v~O. In [1], Lesmes gave sufficient conditions for a subalgebra AcCm(E) to be dense in (Cm(E), z~), when m = l , and E is a real separable Hilbert space. In fact, he proved that (1), (2), (3) (with U=E) and

An extension of Nachbin’s theorem to differentiable functions on Banach spaces with the approximation property