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  "body": "\n## Summary\n\nLet $X_0$ and $X_1$ be two Banach spaces that sit in a common space.\nAn interpolation space for them is a Banach space between them with this property.\nEvery linear map bounded on both $X_0$ and $X_1$ is bounded on it too.\nThe model case is the Riesz–Thorin theorem.\nIt says that the [[Lp spaces|SASP14MA1Z]] with $p$ between $p_0$ and $p_1$ are interpolation spaces for $L^{p_0}$ and $L^{p_1}$.\nThe real method of Lions and Peetre and the complex method of Calderón build interpolation spaces in general.\nInterpolation gives bounds for operators such as the Fourier transform and singular integrals, and it defines fractional Sobolev spaces.\n\n## Explanations\n\n### L1 human\n\nSuppose you know that an operation is well behaved for two extreme ways of measuring size.\nInterpolation lets you conclude that it is also well behaved for the measures of size in between.\nFor example, the Fourier transform is bounded from $L^1$ to $L^\\infty$ and from $L^2$ to $L^2$.\nInterpolation then gives the Hausdorff–Young inequality: it is bounded from $L^p$ to $L^q$ for $1 \\le p \\le 2$ and $1/p + 1/q = 1$.\n",
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