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 $H^2$.