The Bernstein problem calls for necessary and sufficient conditions in order that the set of functions xnk(x), n =0, 1, * * *, form a fundamental set2 in the linear space C0 of functions continuous on (- 0, co), vanishing at ? co, and normed by the maximum. A few simple observations help to reduce this formulation of the problem to a more accessible form. First, if xnk(x) is fundamental, then k(x) can have no zeros. For each element in the closure of the finite linear combinations of {xnk(x) } must inherit any zero of k(x). Secondly xnk(x), n =0, 1, , can be fundamental if and only if XnIk(x) is also fundamental. This is true because a function f(x) in Co is approximable by linear combinations of the xnk(x) if and only if the continuous function3 f(x) sgn k(x) is approximable by the same linear combinations of xn|k(x)1. Consequently it may be assumed that k(x) =I /b(x), where (ii) l~~~~im Xnlb(X) = 0, X > + o00 n = O, 1, *** (iii) ~ ~ ~ ~ ~ 4 b(x) > O,1 - oo (x) which in addition to (ii) and (iii) have the