This note is an improvement of the available methods for getting results on the weighted approximation of continuously differentiable functions. We shall present what we believe to be a simplified version of the reasoning to establish some results concerning weighted approximation of continuously differentiable scalar functions on R" (see [3] ). For references to weighted approximation of continuous scalar functions on R" (see [1] and [3]). Lemma 1 reduces the search for sufficient conditions

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... order that a weight on R be Cm -fundamental to the finding of sufficient conditions for a weight on R to be C-fundamental. Similarly, Lemma 2 reduces the finding of sufficient conditions for a weight on R" to be Cm -fundamental to the search of sufficient conditions for a weight on R to be Cm -fundamental. We then apply Lemmas 1 and 2 together to obtain Propositions 2, 4, and 5. Fix integers n e N, n > 1, and m e N (the case m = ce will be excluded since it follows easily from all m E N ). Let K denote either R or C. Consider the algebras &>(Rn) of all K-valued polynomials on R" and Cm(R") of all continuously m -differentiable K-valued functions on R" . Write N^ = {a e N" : |q| < m} where \a\ = ax + ■ ■ ■ + an if a = (ax, ... , an) e Nn . Let Daf be the ath partial derivative of / € Cm(R") for aeN"m. Denote by 2m(Rn) the subalgebra of Cm(R") of all functions with compact support. A Cm-weight on R" is a family v = {va: a e N^} of upper semicontinuous functions va > 0 on R" . Such a weight v defines the vector space Cmv00(R") of all / £ Cm(R") such that vaDaf tends to zero at infinity for every a e N"m. Set \\f\\v = suv{va(t) ■ \Daf(t)\: t e Rn} to get a seminorm / € C'v^R") t-\\f\\v e R+ for every a e Nnm . The finite family of such seminorms makes Cmvoo(Rn) into a seminormable space, actually a