{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1BXEJXQ2YT"
    },
    {
      "field": "about",
      "target": "2MTN8CQWME"
    },
    {
      "field": "about",
      "target": "7ZR88SCSRF"
    },
    {
      "field": "about",
      "target": "FX5PWKCMBQ"
    },
    {
      "field": "about",
      "target": "NYVHMQJP3E"
    },
    {
      "field": "about",
      "target": "RKKWS7CDNF"
    },
    {
      "field": "uses",
      "target": "1BXEJXQ2YT"
    },
    {
      "field": "uses",
      "target": "2MTN8CQWME"
    },
    {
      "field": "uses",
      "target": "61B8B11Y0Y"
    },
    {
      "field": "uses",
      "target": "7ZR88SCSRF"
    },
    {
      "field": "uses",
      "target": "ARG1F8W9NZ"
    },
    {
      "field": "uses",
      "target": "AS7ZA3P37H"
    },
    {
      "field": "uses",
      "target": "FX5PWKCMBQ"
    },
    {
      "field": "uses",
      "target": "K4S9G3AQBM"
    },
    {
      "field": "uses",
      "target": "NYVHMQJP3E"
    },
    {
      "field": "uses",
      "target": "Q3Y1HBD8NN"
    },
    {
      "field": "uses",
      "target": "RKKWS7CDNF"
    },
    {
      "field": "uses",
      "target": "W35ZP39S4R"
    },
    {
      "field": "defines",
      "target": "1DRW7JP962"
    }
  ],
  "body": "\n## Summary\n\nLet $(X, Y)$ be a [[dual pair|R68HNB5SVR]].\nThe weak topology $\\sigma(X, Y)$ is the coarsest topology on $X$ making every map $x \\mapsto \\langle x, y \\rangle$ continuous.\nFor a topological vector space $E$, the weak topology is $\\sigma(E, E')$, defined by its [[continuous linear forms|4A82S5FVRT]], and the weak-* topology on $E'$ is $\\sigma(E', E)$.\nWeak topologies have fewer open sets than norm topologies, so more sets are compact; this is the content of the [[Banach–Alaoglu theorem|1BXEJXQ2YT]].\n\n## Explanations\n\n### L1 human\n\nIn the weak topology, a sequence converges when every continuous linear measurement of it converges.\nThis is much easier to satisfy than convergence in norm.\nIn $\\ell^2$, the unit vectors $e_1, e_2, e_3, \\ldots$ converge weakly to $0$: for each fixed $y \\in \\ell^2$, the numbers $\\langle e_n, y \\rangle = y_n$ tend to $0$.\nYet $\\norm{e_n} = 1$ for all $n$, so they do not converge to $0$ in norm.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46A20"
    ],
    "name": "Weak topology",
    "same_as": {
      "wikidata": "Q505760",
      "wikipedia": "Weak topology"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weak_topology"
      }
    ],
    "tag": "P6KHVSTPZV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/P6KHVSTPZV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/P6KHVSTPZV.md",
  "msc": [
    "46A20"
  ],
  "name": "Weak topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P6K/P6KHVSTPZV.md",
  "status": null,
  "tag": "P6KHVSTPZV",
  "uri": "m2://P6KHVSTPZV",
  "url": "https://math2.tamnd.com/wiki/weak-topology/",
  "writing": {
    "tag": "P6KHVSTPZV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}