Consider the linear space of all real valued polynomials $P$, equipped with the inner product $\langle f, g\rangle=\int^{\infty}_{-\infty}f(t)g(t)e^{-t^2}dt$
(a) Verify that this is an inner product indeed. (You must also explain why the integral converges.) (b) The standard basis on $P$ consists of all monomials $1, t, t^2,...,t^n...$ Use a Gram-Schmidt process to obtain the first 4 orthogonal polynomials: $p_0$, $p_1$, $p_2$ and $p_4$. The polynomials $p_0, p_1,\cdots, p_n\cdots$ obtained by the Gram-Schmidt process are called Hermite polynomials.