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You want a version of the classical Müntz–Szász theorem for the space $L^2([0,1])$ (which is, incidentally, the case considered initially by Szász). Here is a nice paper on the situation for $L^p([0,1])$ spaces, always for the span of monomials (with real exponents allowed) http://www.math.tamu.edu/~terdelyi/papers-online/Fullmuntz.pdf