In Wikipedia's Hilbert space article on separability, it says:

A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to $ℓ^2$.

I guess orthonomal basis means Hamel basis, since Schauder basis are always countable by definition. But according to this theorem, any Banach space of infinite dimenesion will have uncountable Hamel basis only.

So why does it mention infinite-dimensional separable Hilbert spaces? Isn't that a contradiction?

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    $\begingroup$ No, orthonormal basis means Hilbert basis (which is, in the countable case, also a Schauder basis). A complete normed space cannot have a countable Hamel basis (by Baire's theorem). $\endgroup$ – Daniel Fischer Sep 29 '14 at 21:35
  • $\begingroup$ @DanielFischer Ok, I'll try to understand the difference between Hilbert and Schauder basis. Btw, you say "complete normed space cannot have a countable Hamel basis", that's for the infinite dimensional case, right? $\endgroup$ – jinawee Sep 29 '14 at 21:41
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    $\begingroup$ In that, "countable" means "countably infinite". Yes, finite-dimensional spaces are complete. $\endgroup$ – Daniel Fischer Sep 29 '14 at 21:43

$\ell^2$ itself is a simple example of an infinite-dimensional separable Hilbert space.

The orthonormal basis is just a maximal orthonormal set -- so a Hamel basis of a dense subspace which is, in addition, orthonormal. It can certainly be countable.

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