Interpolation space

Summary

Let $X_0$ and $X_1$ be two Banach spaces that sit in a common space. An interpolation space for them is a Banach space between them with this property. Every linear map bounded on both $X_0$ and $X_1$ is bounded on it too. The model case is the Riesz–Thorin theorem. It says that the Lp spaces with $p$ between $p_0$ and $p_1$ are interpolation spaces for $L^{p_0}$ and $L^{p_1}$. The real method of Lions and Peetre and the complex method of Calderón build interpolation spaces in general. Interpolation gives bounds for operators such as the Fourier transform and singular integrals, and it defines fractional Sobolev spaces.