I have a vector space of polynomials at most second degree with basis $\displaystyle (1,x,x^2)$ with the inner product defined as

$\displaystyle \langle{p,q}\rangle=\int^{10}_{0}p(x)q(x)dx$

and I want to find an orthonormal basis using the Gram-Schmidt process.

I've found that $\displaystyle e_1=1/10$

For $\displaystyle e_2$ I have

$\displaystyle e_2=(x-\langle{x},\frac{1}{10}\rangle\frac{1}{10})/||x-\langle{x},\frac{1}{10}\rangle\frac{1}{10}||$

$\displaystyle e_2=(x-1/2)/||x-1/2||$

$\displaystyle e_2=(x-1/2)/16.90661$

However when I take the inner product between $\displaystyle e_1$ and $\displaystyle e_2$ the result is not zero, but it should be since they're orthogonal. However if I take the inner product with the upper limit at 1 they are zero.

I'm pretty sure I've been applying the Gram-Schmidt process using the inner product I defined above. Does anyone know what I did wrong?