Ring of polynomial functions

In mathematics, the ring of polynomial functions gives a coordinate-free analog of a polynomial ring. It can be motivated as follows. If $S=R[t_{1},\dots ,t_{n}]$, then, as the notation suggests, we can view $t_{i}$ as coordinate functions on $R^{n}$: $t_{i}(x)=x_{i}$ when $x=(x_{1},\dots ,x_{n}).$ This suggests the following: let V be a finite-dimensional vector space over an infinite field k and then let S be the subring generated by the dual space $V^{*}$ of the ring of functions $V\to k$ (or, more succinctly, $S=\operatorname {Sym} (V^{*})$.) If we fix a basis for V and write $t_{i}$ for its dual basis, then S consists of polynomials in $t_{i}$; S can be viewed as a polynomial ring over k. If V is viewed as an algebraic variety, then this S is precisely the coordinate ring of V.