Q: Apply Gram-Schmidt in $L^2[-1,1]$ to the list of functions $\{1,x,x^2,x^3\}$. You do not have to normalize.
I have encountered Gram-Schmidt with vectors:
$$U_1 = V_1$$
$$U_2 = V_2 - \frac{\left<V_2,U_1\right>}{\|U_1\|^2}U_1$$
As well as $L^2$ spaces with functions: $\frac{1}{a-b}\int_b^af(x)dx$
However I am getting stuck on how to connect the two ideas. Would this question require me to treat the list like a vector or should I integrate each member of the list?
Yes, each function in $\{1,x,x^2,x^3\}$ is a vector in $L^2[-1,1]$, remember that $L^2[-1,1]$ is a vector space. To apply the Gram–Schmidt process use $U_1=1,\ U_2=x,\ U_3=x^2,\ U_4=x^3$ and use the inner product in $L^2[-1,1]$:
$$\left<f,g\right>=\int_{[-1,1]}fgd\mu$$