{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "1GAXM9QDCQ"
    }
  ],
  "body": "\n## Summary\n\nThe approximate point spectrum of an operator $T$ is the set of numbers $\\lambda$ that are eigenvalues up to an arbitrarily small error.\nThis means that there are unit vectors $x$ with $Tx - \\lambda x$ as small as one likes.\nIt contains every eigenvalue and the whole continuous spectrum, and also the boundary of the spectrum.\nFor self-adjoint and normal operators on Hilbert spaces it is the whole spectrum.\n\n## Explanations\n\n### L1 human\n\nAn eigenvector is a vector $x$ with $Tx = \\lambda x$ exactly.\nAn approximate eigenvector only needs $Tx$ to be very close to $\\lambda x$, compared with the size of $x$.\nFor the operator that divides the $n$-th entry of a sequence by $n$, the $n$-th unit vector is sent to a vector of length $1/n$.\nSo $0$ is an approximate eigenvalue, although it is not an eigenvalue.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47A10"
    ],
    "name": "Approximate point spectrum",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)"
      }
    ],
    "tag": "N44162Y9XS"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/N44162Y9XS/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/N44162Y9XS.md",
  "msc": [
    "47A10"
  ],
  "name": "Approximate point spectrum",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N44/N44162Y9XS.md",
  "status": null,
  "tag": "N44162Y9XS",
  "uri": "m2://N44162Y9XS",
  "url": "https://math2.tamnd.com/wiki/approximate-point-spectrum/",
  "writing": {
    "tag": "N44162Y9XS",
    "kind": "concept",
    "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
      }
    ]
  }
}