Let I-identity operator, \Pi_N is the orthogonal projection in L_2 onto subspace by the first N eigenfunctions of the Stokes operator in \Omega, \alpha_j denotes the increasing sequence of the eigenvalues for the Stokes operator, c>0 is the some constant not depending on N. We know that \Pi_N v = 0.

How to derive the following inequality using the Poincare inequality? Which form of inequality is used here? |v|{L_2}=|(I-\Pi_N)v|{L_2}\le c\alpha_N^{-1/2}|v|_{H^1}.