The Best Constant for Error in Orthogonal Projection onto Finite-dimensional Subspaces of Abstract Hilbert Spaces

Abstract

Abstract. The prior error evaluation of the Galerkin method for the Poisson equation is expressed by the projection, and the constants have been studied to evaluate the convergence and error of the approximate solution. This paper considers error constants for orthogonal projections on finite-dimensional subspaces of the abstract Hilbert space. By proving the converse of the Aubin-Nitsche technique without restricting compactness or the basis, we show that the constants satisfying two inequalities are equal. Next, it is also shown that the best error constant under the compactness assumption is the smallest eigenvalue of the eigenvalue problem.