{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0PARCKE8WP"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $(Y, d_Y)$ be a [[complete metric space|08EWT7J0SC]].\nLet $C_b(X, Y)$ be the set of [[continuous maps|0SY9ECQ19Q]] $f \\colon X \\to Y$ whose image $f(X)$ is [[bounded|0P953MG05E]] in $Y$.\nFor $f, g \\in C_b(X, Y)$, let $d_\\infty(f, g) = \\sup_{x \\in X} d_Y(f(x), g(x))$, with $d_\\infty(f, g) = 0$ when $X$ is empty.\nThen $(C_b(X, Y), d_\\infty)$ is a complete metric space.\n\n## Remarks\n\nFor $Y = \\mathbb{R}$, the metric $d_\\infty$ is the [[uniform metric|06HE9J7DXF]], and it comes from the [[sup norm|0PARCKE8WP]].\nProof sketch: a Cauchy sequence $(f_n)$ in $d_\\infty$ is Cauchy at each point $x$, so it has a pointwise limit $f(x)$ in $Y$. The convergence is uniform, and a uniform limit of continuous maps is continuous. The limit is also bounded.\nFor a compact space $X$, each continuous map to $\\mathbb{R}$ is bounded, so $C(X, \\mathbb{R})$ is complete with the sup metric.\n\n## Explanations\n\n### L1 human\n\nIf continuous functions get closer and closer to each other uniformly, then they converge uniformly to a function.\nThe key point is that this limit function is again continuous, so the limit stays in the space.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "54C35"
    ],
    "name": "Completeness of the space of bounded continuous functions",
    "role": "theorem",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Uniform_norm"
      },
      {
        "locator": "Examples",
        "url": "https://en.wikipedia.org/wiki/Complete_metric_space"
      }
    ],
    "tag": "09GRCZVR5E",
    "uses": [
      "08EWT7J0SC",
      "0SY9ECQ19Q",
      "06HE9J7DXF"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/09GRCZVR5E/",
  "links": [
    {
      "field": "uses",
      "target": "08EWT7J0SC"
    },
    {
      "field": "uses",
      "target": "0SY9ECQ19Q"
    },
    {
      "field": "uses",
      "target": "06HE9J7DXF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09GRCZVR5E.md",
  "msc": [
    "54C35"
  ],
  "name": "Completeness of the space of bounded continuous functions",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09G/09GRCZVR5E.md",
  "status": null,
  "tag": "09GRCZVR5E",
  "uri": "m2://09GRCZVR5E",
  "url": "https://math2.tamnd.com/wiki/completeness-of-the-space-of-bounded-continuous-functions/",
  "writing": {
    "tag": "09GRCZVR5E",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}