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.
2026-03-31 14:12:20.1774966340
Orthonormal basis for Sobolev Spaces
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As mentioned above, the question is not well-posed in this form since the answer depends not only on the precise space you are considering but also on the norm you are using. Perhaps the following comment might nevertheless be useful. If your space can be identified with (or is defined as) the domain of definition of an unbounded, self-adjoint operator (as many of the useful ones are) and if the latter has discrete spectrum, then its eigenfunctions (suitably normed) form an ONB for the Sobolev space with the corresponding norm. Simple examples are the Laplace operator on the circle or the standard one-dimensional Schrödinger operator on the line. More sophisticated examples are provided by the Laplacian on a compact Riemann manifold or general Schrödinger operators under suitable conditions on the potential function.