I'm trying to read chapter 3.7 from Kreyszig where he talks about Legendre, Hermite, and Laguerre polynomial. Here's how the definitions are -
Consider the space $L^2[0, \infty)$ with inner product $\langle f, g \rangle = \int_0^\infty fg dx$. Define $e_n(t) = e^{-\frac{t}{2}}L_n(t)$ where $L_n(t)$ is the n-th Laguerre polynomial given by $L_0(t) = 1$ and $L_n(t) = \frac{e^t}{n!} \frac{d^n}{dt^n}(t^ne^{-t})$. The claim is that $(e_n)$ is total in $L^2[0, \infty)$.
I came across this answer but it uses generating functions that I don't know. So far we have been taught up to chapter 4 from Kreyszig and up to chapter 5 from Axler's Measure theory. So I was wondering if it can be proven using things I know, or does it require using advanced techniques.