let $V$ be a real vector space. I would like to know what is meant by a "polynomial map" $f:\times^k V\rightarrow \mathbb{R}.$
I suppose the space $\times^k V$ is the cartesian product $V^k$ and therefore a vector space by itself. In algebra textbooks I could found the definition of polynomials $R[X]$ or even $R[X_1,...,X_n]$ for any ring $R$. But in the above situation, $\mathbb{R}$ is the only ring I can recognize.
My background is more in the realm of analysis than in algebra. So I would be very happy if anyone can tell me with my issue in simple language...
Best wishes