Sobolev type norms and basis functions

[ahg187]

Hello everybody,

I am given a "Sobolev type innerproduct"

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

for some \alpha \geq 0 and R 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!

 

[ahg187]

I figured it out!

I showed that the norm induced by \langle \cdot, \cdot \rangle_{\alpha} is indeed equivalent to the standard Sobolev norm \Vert \cdot \Vert_{W^{2,2}} of the Sobolev space W^{2,2} which is of course the (reproducing kernel) Hilbert space H^2.