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

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SUMMARY

The discussion clarifies the differences between H1 and H2 Hilbert spaces, which are specific Sobolev spaces denoted as H^k = W^{k,2}. The "2" indicates the use of the L^2 norm, while "k" signifies the inclusion of derivatives up to the kth order in the function analysis. Specifically, the norm is defined as ||f||^2_{H^k} = ||f||^2_{W^{k,2}} = ∑_{i=0}^k ∫ dt |f^{(i)}|^2, highlighting the importance of derivatives in these spaces.

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amalmirando
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Hilbert Space...

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

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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