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I am given a "Sobolev type innerproduct"

[itex]\langle f,g \rangle_{\alpha} = \langle f,g \rangle_{L^2} + \alpha \langle Rf,Rg \rangle_{L^2}[/itex]

for some [itex]\alpha \geq 0[/itex] and [itex]R[/itex] some differential operator (e.g. the second-derivative operator).

My question is now whether a function space endowed with such an inner-product can be a Hilbert space?

Also I wonder whether there exists an analogue to the spectral theorem for compact and self-adjoint operators on Hilbert spaces also in Sobolev spaces?

Thanks for any input on these questions!

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# Sobolev type norms and basis functions

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