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We present an approach to accelerate real-space electronic structure methods several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient, systematically improvable, discontinuous basis spanning the occupied subspace and project the real-space Hamiltonian onto the span. In calculations on a range of systems, we find that accurate energies and forces are obtained with 8-25 basis functions per atom, reducing the dimension of the associated real-space eigenproblems by 1-3 orders of magnitude. [ABSTRACT FROM AUTHOR]