$L^2(-1,1)$
$f_1(x)=1+x$ , $f_2(x)=2-x $ , $f_3(x)=1$
I know the algorithm and what it does, but it's not clear how to proceed.
I think it would be better to take as "weight function" $f_3(x) $ .
$\hat{f}_1(x)=f_1(x) $
$\hat{f}_2(x)=f_2(x)-<\hat{f}_1,f_2>\hat{f}_1(x) $
I think.
Not sure I understand - if you know the algorithm, why don't you know how to proceed?
Note that $$ \left|f_1(x)\right|^2 = \left<f_1, f_1 \right> = \int_{-1}^1 f_1^2(x)dx = \int_{-1}^1 (1+x)^2dx = 8/3. $$ Indeed, $$ \hat{f}_1(x) = \frac{f_1(x)}{|f_1(x)|} = \frac{f_1(x)}{\sqrt{8/3}} = f_1(x)\sqrt{3/8} $$ and you define $\hat{f}_2(x)$ correctly. Can you compute it, and define and compute $\hat{f}_3(x)$ similarly?