Let $V$ be the vector space of real polynomial $\mathbb{R}[x]$ endowed with the inner product
$\langle f,g \rangle = \displaystyle\int_{-\infty}^{\infty} e^{-|x|}f(x)g(x) \ dx$
By considering the sequence of subspaces $\{V_n\}$ where
$V_n = \{f(x) \in \mathbb{R}[x] : \deg f \leq n \}$
or otherwise, show that there exist unique monic polynomials $\phi_n(x)$ for $n \geq 0$ such that
$\displaystyle\int_{-\infty}^{\infty} e^{-|x|}\phi_n(x)g(x) \ dx = 0$
whenever $\deg g < n$, and find $\phi_n(x)$ for $n = 0,1,2.$
What is the coefficient of $x^{2000}$ in $\phi_{2007}(x)$?
I'm having trouble even knowing where to begin with this question, any help appreciated!