Is the usual sobolov space H^2(R^3) just the Hilbertspace L^2(R^3)?

Question

It is usually said that for a free particle in $R^3$, the free particle Hamiltonian acts on a Hilbert space $L^2(R^3)$. My question is whether "the usual Sobolev space" $H^2(R^3)$ is just the Hilbert space $L^2(R^3)$? Thank you for your help!

Answer 1

No, it the space of all functions in $L^2(R^3)$ which have first and second order partial derivatives (in the distributional sense) in $L^2(R^3)$