What are the key differences between H1 and H2 Hilbert spaces?

[amalmirando]

Hilbert Space...

Can somebody tell me the difference between H1 & H2 hilbert spaces?

 

[tom.stoer]

Can somebody tell me the difference between H1 & H2 hilbert spaces?

These are two special Sobolev spaces, namely

$$H^k = W^{k,2}$$

The "2" refers to the usual norm of the $$L^2$$ Hilbert space. The "k" means that instead of the function $$f(x)$$ one considers in addition its first, second, ..., kth derivative:

$$||f||^2_{H^k} = ||f||^2_{W^{k,2}} = \sum_{i=0}^k\int dt\,|f^{(i)}|^2$$