Let $V,W$ be finite dimensional real vector spaces, and let $(v,w) \not= (x,y)$ be two points in $V \times W$. Is it possible to construct a bilinear map $\alpha: V \times W \to \mathbb{R}$ such that $\alpha (v,w) \not= \alpha (x,y)$? If so, how do I define such a map?
The reason I want to know is that this is the last step in my verification that $\operatorname{Bilin}(V,W;\mathbb{R})^*$ is the tensor product of $V$ and $W$. Any help would be appreciated, thank you.