A method for reducing the order of a matrix inversion required in the linear approximation problem in abstract Hilbert space. The components of the basic vectors of the approximating vectors are identified with the members of an orthonormal sequence in the Hilbert space. Another, less preferred, method of partitioning by block decomposition is briefly presented. Then the preferred method is applied to the problem of the best least-squares approximation to a function x(t) defined over a finite interval under three conditions. In each of these cases, the matrix inverse required is at most of the order of n+1-r.
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