1. Introduction. The convergence of least squares approximations for dual orthogonal series in Hilbert space is established, thus providing a theorem applicable to practically all dual orthogonal series (such as dual trigonometric series, dual Bessel series, etc.) that have appeared in the literature. Our results establish for such dual series the existence of a sequence of functions satisfying in the L 2 norm the dual series relation, with an error tending to zero and, in particular, rigorously justify the calculations in [2] which showed least squares to be a practical approximation procedure for dual trigonometric equations. In fact, the desire to provide a rigorous convergence theorem for [2] motivated this study.Standard definitions and notation for Hilbert space are employed [1]. We denote by R a real, separable, abstract Hilbert space. The subspaces P and Q are orthogonal complements, while P and Q denote respectively the projection operators from R onto P and onto Q, We recall that P+ Q is the identity operator. {#": n = 1,2,...} denotes a complete orthonormal sequence in R, while $ = {*!/"} is defined by ip n = a n P(j> n + b n Q(j) n , where {a n } and {b n } are real, nonnegative sequences. In R the dual orthogonal series problem is this: Given {a n },
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