Let $V=L{ }^2}([0,1], C$ and $\left\langle_{0} ight angle: V \times V ightarrow C, \langle f, E angle=\int_[0] {1} f(x) \overline g(x)} d x$ be an innerproduct on $V$, Let. $90 5_{2), 5_{2}, _{3} ;[0, 1] ightarrow 8$ be $g i venbv-] 5_{}(x)=1, E_{1}x)=x, x_{2} (x)=x^{2}, {3} (x)=x* [3]$ and consider the following subset $S=\left(0, 81), E_{2}, g_{2} ight) \subset V$. Ater applying the Gram. Schmiat process the followg set of vectars $T=\left\ {v_[0], v_{2}, v_{2), v_{3} ight\}$ is an orthonornal set, where $v_{1}, v_{2}, v_{3}$, and $v_{4}$ are given by $$ \begin{array}1) v_{0}=1, v_{2}=\sqrt{3}(2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-5 x+l ight), v_{4}=\sqrt{7\left(20 X^{3}-30 x^{2}+12 x-1 ight)} \\ v_(O)=1, v_{2}=\sqrt{3}(2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-7 x+l ight), v_{4}=\sqrt{7}\left(20 x^{3}-30 x^{2}+12 x-1 ight) W v_{0=1, v_{2}=\sqrt(3)(2x-1), v_{3}=\sqrt{5}\left(6 x^{2}+6 x+1 ight), v_{4}=\sqrt{7}\left(20 **{3}-30 x^{2} +12 x-l ight) W v_{0)=1, v_{2}=\sqrt{3} (2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-5 x-1 ight), v_{4}=\sqrt{7}\left(20 *^{3} - 30 x^(2) +12 X-1 ight) W v_{0 =1, v_{2}=\sqrt{3}(3 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-6 x+1 ight), v_{4}=\sqrt{7}\left(20 x^{3}-30 x^(2) +12 X-1 ight) W v_{0 =1, v_{2}=\sqrt{3} (2 x+1), v_{3}=\sqrt{5}\left(5 x^{2}-6 x+1 ight), v_{4}=\sqrt{7}\left(20 x^{2}-30 x^(2) +12 X-1 ight) W \left. v_{0)=1, v_{2}=\sqrt{3}(2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-5 x+1 ight), v_{4}=\sqrt{7) 20 **{3}-30 x^{2}+12 x-1 ight) v_{0)=1, v_{2}=\sqrt{3} (2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-6 x+1 ight), v_{4}=\sqrt{7}\left(20 **{3}-30 x^{2}-12 x-1 ight) v_(O)=1, v_{2}=\sqrt{3} (2 x-1), v_{3}=\sqrt{5}\left(6 x^{2}-6 x+l ight), v_{4}=\sqrt{7}\left(20 **{3}+30 x^{2}+12 x-1 ight) W v_{0 =1, v_{2}=\sqrt{3}(2 x-1), v_{1}=\sqrt{5}\left(6 x^{2}-6 x+1 ight), v_{4}=\sqrt{7}\left(10 x^{2}-30 x^(2) +12 x-l ight) \end{array) $$ SP SD. 3471