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  "body": "\n## Summary\n\nThe Peetre K-functional $K(t, x)$ measures how well an element $x$ of a sum $X_0 + X_1$ of two Banach spaces splits into pieces in $X_0$ and $X_1$.\nThe size of the second piece carries the weight $t$.\nIt is the basic tool of the real method of [[interpolation spaces|VX1R6FN6PS]].\nIn approximation theory it links smoothness to approximation, since for suitable pairs of spaces it is equivalent to a [[modulus of smoothness|CRPZ1RVNKQ]].\n\n## Explanations\n\n### L1 human\n\nThink of $X_0$ as rough functions measured in a weak norm and $X_1$ as smooth functions measured in a strong norm.\nTo compute $K(t, f)$ you write $f$ as a smooth part plus a rough remainder and pay the size of the remainder plus $t$ times the size of the smooth part.\nFor small $t$ smoothness is cheap, so you can take a smooth part very close to $f$.\nHow fast $K(t, f)$ goes to $0$ as $t \\to 0$ tells you how well $f$ can be approached by smooth functions.\n",
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