Consider $1
Consider the norm: $\left \| f \right \|_{L_{V}^{p}}=(\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx)^{\frac{1}{p}}$
Consider $W\subset L_{V}^{p}(-1,1)$ to be a finite dimensional subspace. For a given $f$ in $L_{V}^{p}(-1,1)$, we define the minimizer $m$ in $W$ such that: $\left \| f(x)-m(x) \right \|_{L_{V}^{p}}= min\left \| f(x)-q(x) \right \|_{L_{V}^{p}}$ for all $q$ in $W$.
We need to show that $m$ satisfies the following:
$\int_{-1}^{1}\left | f(x)-m(x) \right |^{p-2}(f(x)-m(x))q(x)V(x)dx=0$ for all $q\in W$
Any help is ppreciated?