A symmetric-kernel operator is self-adjoint, with real "diagonal" inner products

If K(t,s) is symmetric kernel of the Fredholm integral equation, define the operator K by $K\phi=\int_a^bK(t,s)\phi(s)ds$. Consider: (I) K is self-adjoint operator. (II) Inner product $\langle K\phi,\phi\rangle$ is always real.