Define the notion of an inner product on a finite-dimensional real vector space $V$, and the notion of a self-adjoint linear map $\alpha: V \rightarrow V$.

Suppose that $V$ is the space of real polynomials of degree at most $n$ in a variable $t$. Show that

$$\langle f, g\rangle=\int_{-1}^{1} f(t) g(t) d t$$

is an inner product on $V$, and that the map $\alpha: V \rightarrow V$ :

$$\alpha(f)(t)=\left(1-t^{2}\right) f^{\prime \prime}(t)-2 t f^{\prime}(t)$$

is self-adjoint.