$\langle p,q \rangle = \int_{-1}^1 p^\ast(t)q(t)tdt$ over $\mathbb{P}$
$\langle p,q \rangle = \int_{-1}^1 p^\ast(t)q(t)(t+1)dt$ over $\mathbb{P}$
I believe positivity works for both of them, and I am unsure of how to show the other ways of proving an inner product.
The star denotes complex conjugate.
Positivity condition fails.
Take p,q as 1.
$<p,p>=0$ for first case as
$\int_{-1}^1 t= \frac{t^2}{2}|_{-1}^1$
$t^2=1-1=0$