Show that $P_2\times P_2 \to \mathbb{R}$ is a dot product in $P_2$

Question

I have the relation:

$$\langle p,q\rangle = 2p(−1)q(−1) + p(0)q(0) + 4p(1)q(1)$$

I want to show that this relation is a dot product in $P_2$. I also want an orthonormal base for $P_2$ with relation to this dot product that I got with the canonical base.

Answer 1

Positive -definitness: $$\langle p,p\rangle = 2p(−1)^2 + p(0)^2+ 4p(1)^2\geq 0$$

and $\langle p,p\rangle=0$ iff $p(-1)=p(0)=p(1)=0$, so quadratic polynomial has 3 zeroes so $p(x)=0$ for all $x$.

and so on...