Interpolation space

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach 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 $L p$ and also on a certain space $L q$ , then it is also continuous on the space $L r$ , for any intermediate $r$ between $p$ and $q$ . In other words, $L r$ is a space which is intermediate between $L p$ and $L q$ .

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).

A Banach space $X$ is said to be continuously embedded in a Hausdorff topological vector space $Z$ when $X$ is a linear subspace of $Z$ such that the inclusion map from $X$ into $Z$ is continuous. A compatible couple $(X 0, X 1)$ of Banach spaces consists of two Banach spaces $X 0$ and $X 1$ that are continuously embedded in the same Hausdorff topological vector space $Z$ . The embedding in a linear space $Z$ allows to consider the two linear subspaces

...
Wikipedia