{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$.\nIf $X$ is [[uniformly convex|BEAMC5011M]], then $X$ is [[reflexive|NN8RWPC8WG]].\n\n## Remarks\n\nLet $J \\colon X \\to X^{**}$ be the canonical embedding, and let $\\xi \\in X^{**}$ with $\\norm{\\xi} = 1$.\nBy the [[Goldstine theorem|K4S9G3AQBM]], $\\xi$ is a weak* limit of a net $J(x_\\alpha)$ with $\\norm{x_\\alpha} \\le 1$.\nGiven $\\varepsilon \u003e 0$, let $\\delta$ be as in the definition of uniform convexity, and choose $f \\in X^*$ of norm $1$ with $\\operatorname{Re} \\xi(f) \u003e 1 - \\delta$.\nFor late indices $\\alpha, \\beta$ one has $\\operatorname{Re} f(x_\\alpha + x_\\beta) \u003e 2 - 2\\delta$, so $\\norm{(x_\\alpha + x_\\beta)/2} \u003e 1 - \\delta$, and uniform convexity gives $\\norm{x_\\alpha - x_\\beta} \u003c \\varepsilon$.\nThis shows that $\\xi$ is within any given $\\varepsilon$ of the closed set $J(X)$, so $\\xi \\in J(X)$.\n\nExample: $L^p$ spaces for $1 \u003c p \u003c \\infty$ are uniformly convex by [[Clarkson's inequalities|1V11QM5C07]], so they are reflexive.\nCompleteness is needed.\nThe space of finitely supported sequences with the norm of $\\ell^2$ is an inner product space, hence uniformly convex, but it is not complete and therefore not reflexive.\nThe converse fails: there are reflexive spaces with no equivalent uniformly convex norm.\n\n## Explanations\n\n### L1 human\n\nUniform convexity is a geometric condition on the shape of the unit ball: it is round in a uniform way.\nReflexivity is an abstract condition about the double dual.\nThe theorem says that the geometric condition is enough to give the abstract one.\n\n### L2 human\n\nThe theorem is the usual quick route to the reflexivity of $L^p$ for $1 \u003c p \u003c \\infty$, without computing the dual twice.\nIn a uniformly convex space, weak convergence plus convergence of norms implies norm convergence, and closed convex sets have unique nearest points.\nThese facts and reflexivity make such spaces convenient in variational problems.\nBeing reflexive is invariant under isomorphism, so every Banach space isomorphic to a uniformly convex one is reflexive.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "BEAMC5011M",
      "NN8RWPC8WG"
    ],
    "kind": "statement",
    "msc": [
      "46B10",
      "46B20"
    ],
    "name": "Milman–Pettis theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6860180",
      "wikipedia": "Milman–Pettis theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Milman%E2%80%93Pettis_theorem"
      }
    ],
    "tag": "AJ5RG9XQYZ",
    "uses": [
      "06GZ956657",
      "BEAMC5011M",
      "NN8RWPC8WG"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/AJ5RG9XQYZ/",
  "links": [
    {
      "field": "about",
      "target": "BEAMC5011M"
    },
    {
      "field": "about",
      "target": "NN8RWPC8WG"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "BEAMC5011M"
    },
    {
      "field": "uses",
      "target": "NN8RWPC8WG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AJ5RG9XQYZ.md",
  "msc": [
    "46B10",
    "46B20"
  ],
  "name": "Milman–Pettis theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AJ5/AJ5RG9XQYZ.md",
  "status": null,
  "tag": "AJ5RG9XQYZ",
  "uri": "m2://AJ5RG9XQYZ",
  "url": "https://math2.tamnd.com/wiki/milman-pettis-theorem/",
  "writing": {
    "tag": "AJ5RG9XQYZ",
    "kind": "statement",
    "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
      }
    ]
  }
}