Green function for periodic boundary value problem of $2M$-th order ordinary differential equation is found by symmetric orthogonalization method under a suitable solvability condition. As an application, the best constants and the best functions of the Sobolev inequalities in a certain series of Hilbert spaces are found and expressed by means of the well-known Bernoulli polynomials. This result has clarified the variational meaning of the special values $\ \zeta(2M)\ (M=1,2,3,\cdots)\ $ of Riemann zeta function $\zeta(z)$.