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Here is a quote from http://en.m.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension (accessed: Nov. 22, 2013) :

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space.

But it seems to me that for a lot of quantum systems this is not so. For example, the one dimensional harmonic oscillator has a countable basis of energy eigenstates but an uncountable basis of position eigenstates (whose wave functions are delta functions).

So where am I wrong? I assume the answer has to do with the fact that delta functions are not actually part of the Hilbert space. Is that it?

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As you and Christoph already pointed out, the difference comes from the fact that these contiuous "bases" do not belong to the respective Hilbert space themselves. This is why they are not actually bases at all in the general sense.

Rather, they are useful mathematical tools to expand any states actually belonging to the Hilbert space. As they obey certain orthonormality and completeness conditions they are also referred to as "continuous bases" (rough summary of section 2.1.3 of "Quantum Mechanics" by Claude Cohen-Tannoudji).

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