For all $n\ge 1$, let $(T_k)_{0\le k\le n}$ be the chebychev polynomials of degree $0 ,..,n$.
I need to prove, using the dot product $(P\mid Q)=\int_{0}^{\pi} P(\cos t)Q(\cos t)\, \text{d}t$,
that the family of vectors $\left(\displaystyle\frac{T_k}{\| T_k \|}\right)_{0\le k \le n}$ is the result of gram - schmidt applied to the canonic base $(1,...,X^n)$. I have already proven that $\|T_k\| = \displaystyle\frac{\sqrt\pi}{2}$ for all $k\ge 0$. Now i get stucked on how to proceed, because the caculations are heavy and i have no idea what approach to take. I tried using the recursive relation satisfied by chebychev polynomials, to no avail. Help would be greatly appreciated.