Spectral theorem for compact selfadjoint operators
Suppose $T \in \mathcal{K}(H)$ is selfadjoint. Then there exists a system of orthonormal vectors $e_1,e_2,$... of eigenvectors of $T$ and corresponding eigenvalues $\lambda_1,\lambda_2,$... such that $|\lambda_1|\geq|\lambda_2|\geq|\lambda_3| \geq ...$
$Tx=\sum_{k=1}^{\infty}\lambda \langle x,e_k \rangle$ forall $x \in H $
Now I want to try this theorem on an example:
Let $Tf(x):=\int_0^1(2xy-x-y+1)f(y)dy$
My Problem now is the question of how to calculate the eigenvectors and eigenvalues of $T$.