For any real number, as well as real numbers, we can define Sobolev space. When it is a positive integer, Sobolev space consists of locally integrable functions, where it is satisfied that for any multiple exponent, it exists and belongs to.

Sobolev space is a normed linear space, and it is a Banach space under the following norms:

If space is often recorded as, we use it to represent space because Sobolev space at this time is Hilbert space. When it is non-integer, Sobolev space can be defined by Fourier transform:

The norm of a function is