Let $P$ and $Q$ be two polynomials in $\mathbb{R}[X]$ and let $$\langle P,Q\rangle =\int _{-\infty}^{+\infty}P(x)Q(x)f(x)dx$$ with $f(x) = \frac{1}{\sqrt{2\pi}} \exp(-x^2/2)$.
I would like to show that $\langle P,Q\rangle$ is well-defined. Indeed, I have to show that $x\mapsto P(x)Q(x)f(x)$ is integrable function on $\mathbb{R}$.
Let $R(x)=P(x)Q(x)=\sum_{k=0}^{d}a_{k}X^{k}$.
Neighbourhood of $+\infty$
$$P(x)Q(x)f(x)\sim a_{d}X^{d}e^{\frac{-X^2}{2}}$$
Note that $X^{2}a_{d}X^{d}e^{\frac{-X^2}{2}} \to 0$, then ${\displaystyle a_{d}X^{d}e^{\frac{-X^2}{2}}=o\left(\dfrac{1}{X^2}\right)}$ and we know that is $x \mapsto \dfrac{1}{x^2}$ is integrable function on $(1,+\infty)$
Thus $x\mapsto P(x)Q(x)f(x)$ is integrable function on $(1,+\infty)$
Neighbourhood of $-\infty$
I'm stuck here
For the same reason that $X^2a_dX^de^{-X^2/2}\to0$ when $X\to+\infty$ it is true that $X^2|a_dX^d|e^{-X^2/2}\to0$ when $X\to+\infty$, and so you get that $X^2a_dX^de^{-X^2/2}\to0$ when $X\to\pm\infty$.