I want to implement the Gram Schimdt procedure to the vector space of polynomials of degree up to 5, i.e. I want to find an orthogonal basis from the set of vectors $v=(1,x,x^2,x^3,x^4,x^5)$. The inner product is defined as $$ \langle f,g\rangle=\int_{-\pi}^{\pi}f(x)g(x)dx. $$ The Gram Schimdt process is to first find $u$ such that \begin{align} &u_1=v_1\\ &u_i=v_i -\sum_{j=1}^{i-1}\frac{\langle u_j,v_k\rangle}{\langle u_j,u_j\rangle}u_j,i\ge 1 \end{align}\begin{align} &u_1=v_1\\ &u_i=v_i -\sum_{j=1}^{i-1}\frac{\langle u_j,v_i\rangle}{\langle u_j,u_j\rangle}u_j,i\ge 1 \end{align} and then the orthogonal basis is found by letting $e_i=u_i/\|u_i\|$. I wrote the following code to find $u$, but it went to endless loop. I am an amatur in mathematica and don't know where I am wrong. Thanks for any help.
v = Function[x, Evaluate[x^#]] & /@ Range[0, 5];
u = v;
For[i = 2, i < 7, i++,
u[[i]] = Function[x,
v[[i]][x] -
Sum[Integrate[v[[i]][x]*u[[j]][x], {x, -Pi, Pi}]/
Integrate[u[[j]][x]^2, {x, -Pi, Pi}]*u[[j]][x], {j, 1, i - 1}]
]
]
I am reading Linear Algebra Done Right P108 and want to implement this by hand to deepen my understanding of this topic and I also want to learn more about mathematica functions. Thanks for any help.