  For conforming finite element approximations of the Laplacian eigenfunctions,
a fully computable guaranteed error bound in the $L^2$ norm sense is proposed.
The bound is based on the a priori error estimate for the Galerkin projection
of the conforming finite element method, and has an optimal speed of
convergence for the eigenfunctions with the worst regularity. The resulting
error estimate bounds the distance of spaces of exact and approximate
eigenfunctions and, hence, is robust even in the case of multiple and tightly
clustered eigenvalues. The accuracy of the proposed bound is illustrated by
numerical examples. The demonstration code is available at
https://ganjin.online/xfliu/EigenfunctionEstimation4FEM .
