Exercise 7.C.14

Proof. In the exercise, $T$ was already shown to be self-adjoint. So $-T$ is also self-adjoint. Note that $1,\cos <!--\; FUNCTION\; APPLICATION\; -->x,\cos <!--\; FUNCTION\; APPLICATION\; -->2x,\dots <!--\; FUNCTION\; APPLICATION\; -->,\cos <!--\; FUNCTION\; APPLICATION\; -->\mathit{nx},\sin <!--\; FUNCTION\; APPLICATION\; -->x,\sin <!--\; FUNCTION\; APPLICATION\; -->2x,\dots <!--\; FUNCTION\; APPLICATION\; -->,\sin <!--\; FUNCTION\; APPLICATION\; -->\mathit{nx}$ is a basis of $V$ consisting of eigenvectors of $-T$ whose corresponding eigenvalues are all nonnegative. Thus by 7.35 $-T$ is positive. □