[Math] How to show that the dot product is a linear functional

Question

How can I show that the dot product is a linear functional?

If I am given a function I have to show that $w(ax_1+bx_2) = aw(x_1) + bw(x_2)$ for it to be a linear functional. I am having trouble seeing how this works for the dot product.

Answer 1

It's not that $\mathbf x\cdot\mathbf y$ is a linear functional. Rather, let $\mathbf x$ be a fixed vector in $\mathbb R$. Then the map $f(\mathbf y)=\mathbf x\cdot\mathbf y$ is a linear functional.