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.
Let and denote the set of smooth functions on with compact support.
Definition 1. Let and be a sequence in We say converges to denoted if
- for all for a fixed compact
- uniformly on
By we mean
is a vector space, and the above notion of convergence allows us to define closed sets and hence turns into a topological vector space called the space of test functions of and we denote it
Definition 2. A distribution is a continuous linear functional on We will denote the set of distributions on by –noting that we are only considering continuous linear functionals.
We say a sequence of distributions converges to the distribution denoted if for all This gives us a topological vector space structure on as well.
Definition 3. Let be a distribution and We define the -th distributional (weak) derivative of by
where We may also write
We will omit if The gradient of is simply
Let us define the space as the space of functions such that
for all compact We define strong convergence on as convergence in and weak convergence on as weak (pointwise) convergence in for all compact
Let then clearly
Let and and define
This integral is finite since both functions are integrable (use integration by parts and establish bounds). Also if uniformly, then
where which gives us the continuity of So We also have
 Lieb, Elliott and Michael Loss. Analysis. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.