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?(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Hilbert space, orthonormal basis

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