A Hilbert space is separable if and only if it admits a countable orthonormal basis

Asked Sep 19 '18 at 15:56

Active Sep 19 '18 at 15:56

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According to Wikipedia:

A Hilbert space is separable if and only if it admits a countable orthonormal basis

While this statement seems very reasonable, it is not clear to me how one would go about proving it.

asked Sep 19 '18 at 15:56

Dave Shulman

The Gram-Schmidt procedure. – Angina Seng Sep 19 '18 at 15:57
1

Start with a countable dense subset. Use Gram-Schmidt, throwing out dependent elements as you go. Argue that the resulting countable orthonormal subset has a closed span equal to the whole space. – Disintegrating By Parts Sep 19 '18 at 16:20

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