Does there exist a total orthonormal set in a Hilbert space which is not a basis? In a separable Hilbert space every total orthonormal set is a basis. What if the Hilbert space is not separable?
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No, there is no such orthonormal set, countable or uncountable. – Kavi Rama Murthy May 18 '22 at 09:04
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If total means maximal orthonormal set, then it is a basis. – Ryszard Szwarc May 18 '22 at 09:04
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so does it mean that every total orthonormal set is a basis irrespective of whether the space is separable or not? – Naman May 18 '22 at 10:06
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Any orthonormal set of vectors is independent. If "total" means that there is no other vector normal to all of the given vectors then they also span the space so are a basis.
George Ivey
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u mean "complete" orthonormal set?...total set is closure of span equals the space – Naman May 19 '22 at 05:16