If I have a potential inner product space over P2, where
$\left< p, q\right> = p(0)q(0)$
How do I determine whether or not it is an inner product space?
Using the four axioms I have:
1) $\left< p, q\right> = p(0)q(0) = q(0)p(0) = \left< q, p\right>$
2) $c\left< p, q\right> = cp(0)q(0) = \left< cp,q\right>$
3) $\left< p,q+w\right> = p(0)(q(0)+w(0)) = p(0)q(0) + p(0)w(0) = \left< p,q\right> + \left< p,w\right>$
4) $~$a)$ \left< p,p\right> = p(0)p(0) = (p(0))^2 > 0$
$\quad$ b) $\left< p,p\right> = p(0)p(0) = (p(0))^2 = 0 \iff p(0) = 0$
I believe I have made an error - I think it might be in axiom 4.
Thank you
$p(0)=0 $ does not imply that $p$ is zero for example $p(x)=x\Rightarrow p(0)=0$ but $p$ is not identically zero.
But you need $\left< p,p\right>=0$ iff $p\equiv0$