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Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis of such a Sobolev space.

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    First of all, there are infinitely many orthonormal bases for any Hilbert space of dimension $>1$. More importantly, finding a basis depends very much on the domain in question and about which Sobolev space you are looking at, i.e., how many derivatives you consider.2012-10-16

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