 In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability.Many methods were designed to generate such spaces of functions, including the Fourier transform, complex interpolation, 
real interpolation, 
as well as other tools (see e.g.fractional derivative).