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Published online by Cambridge University Press: 31 October 2024
A sequence of norm-one elements of a Hilbert space that are mutually orthogonal is said to form an orthonormal sequence. If, additionally, such a sequence spans the entire space, it is said to be complete. As it turns out, if in a Hilbert space there is a complete orthonormal sequence, this space is indistinguishable from the space of square summable sequences. In particular, perhaps contrary to our misleading intuition saying that there are many more square integrable functions than there are square summable sequences, the space of the former is as large as (in fact much the same as) the space of the latter. We will see one important consequence of this stunning result in the next chapter.
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