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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 Dotsenko Jun 8 '11 at 21:23
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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 Yuan Jun 8 '11 at 21:33
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closed as no longer relevant by Bruce Westbury, Theo Johnson-Freyd, Ben Webster Jun 8 '11 at 22:55

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