Consider V a vector space and the symmetric algebra $S(V^*)$
is it possible to define the polynomial on $V$, $R[V]$ canonically ?
I.e. without a use of base ?
And show this is isomorphic to the symmetric algebra ?

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Well, what prevents you from defining polynomial functions on $V$ as $S(V^*)$? It clearly makes sense: $V^*$ consists of all linear functions, and multiplying them to to create polynomials is a natural thing. Or are you worried about the possible side effects in positive characteristic?
– Vladimir DotsenkoJun 8 '11 at 21:23

1

Polynomials on $V$ are precisely elements of $S(V^{\ast})$ by the universal property of the symmetric algebra. There are no characteristic issues here.
– Qiaochu YuanJun 8 '11 at 21:33