Inner product on $\mathbb{Q}$

Question

I have a short question. Is there some inner product on $\mathbb{Q}$?

By Inner product a I mean, a function $\langle, \rangle:V\times V\to\mathbb{Q}$ which is linear in its first argument, positive definitess and symmetric, where $V$ is a vector space over $\mathbb{Q}$.

Answer 1

Yes, for example, if $V=\mathbb{Q}^n$ then

$$\langle (x_1,...,x_n),(y_1,...,y_n)\rangle=\sum_{i=1}^n x_iy_i$$

is symmetric, bilinear and positive definite.

Up to a $\mathbb{Q}$-linear change of coordinates, every inner product on $\mathbb{Q}^n$ is equivalent to one of the form

$$\langle (x_1,...,x_n),(y_1,...,y_n)\rangle=\sum_{i=1}^n a_ix_iy_i$$ where $a_i$ are non-zero squarefree integers.