In the field of mathematical analysis
, an interpolation space
is a space which lies "in between" two other spaces. The main applications are in Sobolev spaces
, where spaces of functions that have a noninteger number of derivatives
are interpolated from the spaces of functions with integer number of derivatives.
The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem. In simple terms, if a linear function is continuous on a certain space and also on a certain space , then it is also continuous on the space , for any intermediate r between p and q. In other words, is a space which is intermediate, or between and .
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).
In order to discuss some of the main results of the theory, it is necessary for the reader to have some familiarity with the theory of Banach spaces. In this article, we are interested in the following situation. X and Z are Banach spaces, and X is a subset of Z, but the norm of X is not the same as the one of Z. An example of this can be obtained by taking X to be the Sobolev space and by taking Z to be . We say that X is continuously included in Z when we have there is a constant