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.
History 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 Lp and also on a certain space Lq, then it is also continuous on the space Lr, for any intermediate r between p and q. In other words, Lr is a space which is intermediate between Lp and Lq. 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).
The setting of interpolation 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 (X0, X1) of Banach spaces consists of two Banach spaces X0 and X1 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
X 0 ∩ X 1 {\displaystyle X_{0}\cap X_{1}}
and
X 0 + X 1 = { z ∈ Z : z = x 0 + x 1 , x 0 ∈ X 0 , x 1 ∈ X 1 } . {\displaystyle X_{0}+X_{1}=\left\{z\in Z:z=x_{0}+x_{1},\ x_{0}\in X_{0},\,x_{1}\in X_{1}\right\}.}
Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of X0 and X1. It depends in an essential way from the specific relative position that X0 and X1 occupy in a larger space Z. One can define norms on X0 ∩ X1 and X0 + X1 by
‖ x ‖ X 0 ∩ X 1 := max ( ‖ x ‖ X 0 , ‖ x ‖ X 1 ) , {\displaystyle \|x\|_{X_{0}\cap X_{1}}:=\max \left(\left\|x\right\|_{X_{0}},\left\|x\right\|_{X_{1}}\right),}
‖ x ‖ X 0 + X 1 := inf { ‖ x 0 ‖ X 0 + ‖ x 1 ‖ X 1 : x = x 0 + x 1 , x 0 ∈ X 0 , x 1 ∈ X 1 } . {\displaystyle \|x\|_{X_{0}+X_{1}}:=\inf \left\{\left\|x_{0}\right\|_{X_{0}}+\left\|x_{1}\right\|_{X_{1}}\ :\ x=x_{0}+x_{1},\;x_{0}\in X_{0},\;x_{1}\in X_{1}\right\}.}
Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:
… excerpt ends here. Continue reading the full article.
