Recall for a function for we define the Fourier transform of by
and the inverse Fourier transform by
If then a.e..
Now if and with we say is the weak derivative of provided
The Sobolev space is then defined as the set of all functions such that for all These become Banach spaces under the norm
and Hilbert when whence we denote
For we can say that consists of such that
(using Fourier multipliers).
Bessel Potential Spaces:
We can generalize with and and define the Bessel potential space as all such that Note the fractional power of the Laplacian is defined by virtue of the fact that is dense in , so we say
These spaces satisfy our desire of being Banach (and Hilbert when ).
There is an alternative approach. Recall that the Holder space is defined as all functions such that
That is, it is the set of functions on which are and whose -th partial derivatives are bounded and Holder continuous of degree These spaces are Banach under the above norm. We can generalize the Sobolev spaces to incorporate similar properties. Let us define the Slobodeckij norm for with and by
The corresponding Sobolev-Slobodeckij space is defined as all functions such that
where This becomes a Banach space under the norm
 http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)
 Lieb, Elliot and Michael Loss. Analysis. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.
 Evans, Lawrence. Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.