My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously it was talking about orthonormal bases in Hilbert spaces and the idea of maximiality: <g,e_k> = 0 for al k => g=0 (definition of maximality) Why is it we use this definition to characterize and orthonormal basis (e_k) and not that H=span(e_k) and how does it relate to the quote above?