Here's a puzzling fact mentioned in my quantum mechanics text: the eigenvectors of a hermitian transformation span the vector space. The text doesn't mention what the proof of this is, but he does say that it doesn't carry over to infinite-dimensional vector space (such as the Hilbert function space in which wave functions reside).
When I have a little free time, I'm going to look up a) what is the proof, and b) why doesn't it work for infinite-dimensional vector spaces?
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