{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X_0$ and $X_1$ be [[Banach spaces|06GZ956657]] that are both continuously embedded in a Hausdorff topological vector space $Z$.\nThe pair $(X_0, X_1)$ is a *compatible couple*.\nThe spaces $X_0 \\cap X_1$ and $X_0 + X_1$ are Banach spaces with the norms\n$$\\norm{x}_\\cap = \\max(\\norm{x}_{X_0}, \\norm{x}_{X_1}), \\qquad \\norm{x}_+ = \\inf \\set{\\norm{x_0}_{X_0} + \\norm{x_1}_{X_1} : x = x_0 + x_1}.$$\nA Banach space $X$ is an *intermediate space* if $X_0 \\cap X_1 \\subseteq X \\subseteq X_0 + X_1$, with both inclusions continuous.\nAn intermediate space $X$ is an *interpolation space* if the following holds.\nEvery linear map $T \\colon X_0 + X_1 \\to X_0 + X_1$ that restricts to [[bounded linear maps|03TM135G0N]] $X_0 \\to X_0$ and $X_1 \\to X_1$ maps $X$ into $X$, and the restriction $T \\colon X \\to X$ is bounded.\n\n## Remarks\n\nExample: $X_0 + X_1$ is an interpolation space.\nIf $x = x_0 + x_1$, then $T x = T x_0 + T x_1$ with $T x_0 \\in X_0$ and $T x_1 \\in X_1$.\nSo $\\norm{T x}_+ \\le \\max(\\norm{T}_{X_0}, \\norm{T}_{X_1}) \\, \\norm{x}_+$.\n\nNon-example: take $X_0 = \\ell^1$ and $X_1 = \\ell^\\infty$, and let $X = \\ell^1 \\oplus \\R \\mathbf{1}$, where $\\mathbf{1} = (1, 1, 1, \\dots)$.\nWith the norm $\\norm{y + c \\mathbf{1}} = \\norm{y}_1 + \\abs{c}$, the space $X$ is intermediate.\nThe map $T(x_1, x_2, x_3, \\dots) = (x_1, 0, x_2, 0, x_3, 0, \\dots)$ has norm $1$ on $\\ell^1$ and on $\\ell^\\infty$.\nBut $T \\mathbf{1} = (1, 0, 1, 0, \\dots)$ is not in $X$, since it differs from every multiple of $\\mathbf{1}$ by a sequence that is not summable.\n\nBy the Riesz–Thorin theorem, for a measure space and $1 \\le p_0 \u003c p \u003c p_1 \\le \\infty$, the space $L^p$ is an interpolation space for the couple $(L^{p_0}, L^{p_1})$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "VX1R6FN6PS",
    "kind": "definition",
    "msc": [
      "46B70"
    ],
    "name": "Interpolation space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Interpolation_space"
      }
    ],
    "tag": "V60F20N6JT",
    "uses": [
      "06GZ956657",
      "03TM135G0N"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/V60F20N6JT/",
  "links": [
    {
      "field": "defines",
      "target": "VX1R6FN6PS"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "03TM135G0N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/V60F20N6JT.md",
  "msc": [
    "46B70"
  ],
  "name": "Interpolation space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V60/V60F20N6JT.md",
  "status": null,
  "tag": "V60F20N6JT",
  "uri": "m2://V60F20N6JT",
  "url": "https://math2.tamnd.com/wiki/interpolation-space-definition/",
  "writing": {
    "tag": "V60F20N6JT",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}