There are many inner products for polynomials wrt different weights, such as $\int uv$ and $\int uv\frac{1}{\sqrt{1-x^2}}$. My question is do these weights have actual meanings in applied context?
After reading the comments, I see that when we have a basis (say chebyshev polynomials), they are orthogonal wrt some weight, and that makes $\int p_ip_jw=\delta_{ij}$. But I think $\int p_ip_jw$ wouldn't be something that we care about? So I still don't quite understand what exactly is simplified.
I guess I am looking for an example where we have motivation to make chebyshev polynomials orthogonal and hence choose to work with the corresponding inner product.