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?
2026-03-26 09:38:35.1774517915
Total orthonormal set which is not a basis
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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.