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  "body": "\n## Definition\n\nLet $(X_0, X_1)$ be a compatible couple of [[Banach spaces|06GZ956657]].\nSo both are continuously embedded in a common Hausdorff topological vector space, and the sum $X_0 + X_1$ is defined.\nFor $x \\in X_0 + X_1$ and $t \u003e 0$, the *Peetre K-functional* is the [[infimum|04KFTA34C5]]\n$$K(t, x) = K(t, x; X_0, X_1) = \\inf \\set{ \\norm{x_0}_{X_0} + t \\norm{x_1}_{X_1} : x = x_0 + x_1, \\ x_0 \\in X_0, \\ x_1 \\in X_1 }.$$\n\n## Remarks\n\nExample: take $X_0 = X_1 = \\R$ with the absolute value.\nFor a split $x = x_0 + x_1$ the cost is $\\abs{x_0} + t \\abs{x_1} \\ge \\min(1, t) (\\abs{x_0} + \\abs{x_1}) \\ge \\min(1, t) \\abs{x}$, and putting all of $x$ in the cheaper space attains this, so $K(t, x) = \\min(1, t) \\abs{x}$.\n\nNon-example: the K-functional is only defined on $X_0 + X_1$.\nFor $X_0 = X_1 = \\ell^1$ inside the space of all sequences, the sequence $(1, 1, 1, \\dots)$ has no split at all, so $K(t, \\cdot)$ is not defined there.\n\nFor each $x$, the function $t \\mapsto K(t, x)$ is nondecreasing and concave, and $K(1, x)$ is the norm of $x$ in $X_0 + X_1$.\nFor $0 \u003c \\theta \u003c 1$ and $1 \\le q \\le \\infty$, the real [[interpolation space|VX1R6FN6PS]] $(X_0, X_1)_{\\theta, q}$ consists of the $x$ for which $t \\mapsto t^{-\\theta} K(t, x)$ lies in $L^q((0, \\infty), dt / t)$.\nIn approximation theory one takes $X_0 = L^p[a, b]$ and $X_1$ the Sobolev space $W^{k, p}[a, b]$, measured by the seminorm $\\norm{g^{(k)}}_p$.\nThen the quantity $K(t^k, f)$ is bounded above and below by constant multiples of the [[modulus of smoothness|CRPZ1RVNKQ]] $\\omega_k(f, t)_p$ for small $t$.\n",
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