The vector space $ \mathbb{R} [X]_p$ of real-valued polynomial of rank $< p$ is provided with the scalar product $$ \langle f,g\rangle := \int_{-1}^1 f(x)g(x) dx.$$
To Show is that the linear Transformation $L: \mathbb{R} [X]_p \to \mathbb{R} [X]_p, f \to [(1-X^2)f']' $ is self-adjoint regarding $\langle ,\rangle $, and also determine the Eigenvalues and eigenvectors.
I know how to Show self-adjointed matrices..but i have no idea how to do this Task. Please explain it to me.
The adjoint of a function $f$ is the function $g$ such that for all vectors $x, y$:
$$ \langle x, f(y) \rangle = \langle g(x), y \rangle.$$ Thus, you must show that for all polynomials $f, g \in \mathbb{R}[X]_{\leq p}$: $$ \int_{-1}^1 f(x) [(1-x^2)g'(x)]'dx=\int_{-1}^1 [(1-x^2)f'(x)]'g(x)dx.$$ Alternatively, you could transform the question into one of matrices, calculating the matrix for the map $L$ after choosing a basis for $\mathbb{R}[X]_{\leq p}$.