Linear vs. bilinear I'm tripping over something elementary:
Suppose $f:\mathbb{R^2}\rightarrow X$ is linear, then $f(x+y)=f(x)+f(y)$ for all vectors $x$ and $y$. Now suppose that $f$ is also bilinear and in particular linear in the first argument, then
$$f\Big((a+b,c)\Big)= f\Big((a,c)\Big)+f\Big((b,c)\Big)$$
But now setting $x=(a,0)$ and $y=(b,c)$ contradicts linearity of f.

A (non-trivial) linear function from $\mathbb{R}^n$ for $n > 1$ will never be a multilinear function (bilinear when $n=2$) if it is a linear function.  Instead, a multilinear function is a linear function from a tensor product.  So a bilinear function on $\mathbb{R}$ is a linear function $\mathbb{R}\otimes\mathbb{R} \to X$, but $\mathbb{R}$ is the unit for the tensor product, so $\mathbb{R}\otimes\mathbb{R} \cong \mathbb{R}$ which is very much not $\mathbb{R}^2$.
