What is a Sobolev space and Sobolev norm 1 and 2 etc.? Where is it used to? In Quantum Chemistry  for description of many body systems we usually use approximate wave functions and we require that this wave function has to be extended in a complete  basis set. And the completness in this case means completness  in euclidien norm. Why we need completness by Sobolev norm? Is it connected somehow with differential nature of kinetic energy operator? Elaborate in details.