{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{S}$ be the [[Schwartz space|CG9JEATW3Q]], and let $\\mathcal{S}'$ be the space of its [[tempered distributions|AES2T5JPFP]], with the topology of convergence against each element of $\\mathcal{S}$.\n\nLet $q$ be a continuous [[seminorm|FP5GVCN8GV]] on $\\mathcal{S}$.\n\nThe **attainable null space** of $q$ is the set\n\n$$\\mathcal{N}_q = \\{\\, p \\in \\mathcal{S}' : p = \\lim_{k} \\chi_k \\text{ in } \\mathcal{S}' \\text{ for some sequence } (\\chi_k) \\subset \\mathcal{S} \\text{ with } q(\\chi_k) \\to 0 \\,\\}.$$\n\nLet $\\mathrm{B} \\colon \\mathcal{S} \\to \\mathcal{S}'$ be a positive-semi-definite operator. This means that $\\langle \\mathrm{B}\\varphi, \\varphi \\rangle$ is real and nonnegative for every $\\varphi \\in \\mathcal{S}$.\n\nThe seminorm induced by $\\mathrm{B}$ is $|\\varphi|_{\\mathrm{B}} = \\langle \\mathrm{B}\\varphi, \\varphi \\rangle^{1/2}$.\n\nFor this case, $\\mathcal{N}_{\\mathrm{B}}$ denotes the attainable null space of $|\\cdot|_{\\mathrm{B}}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "RK2N3QP4FQ",
    "history": {
      "notes": "The source is arXiv:2610.14906v1 (2026-10), by Michael Unser. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it."
    },
    "inventory": [
      "6YTCGV56SP"
    ],
    "kind": "definition",
    "msc": [
      "46-XX"
    ],
    "name": "Attainable null space",
    "sources": [
      {
        "arxiv": "2610.14906v1",
        "licence": "cc-by",
        "locator": "Definition 3.1",
        "mention": "ax://paper/2610.14906#2P92",
        "note": "the paper that states the item (corpus arxiv AX9.2)"
      }
    ],
    "tag": "Y4WACJKC3S",
    "uses": [
      "FP5GVCN8GV",
      "CG9JEATW3Q",
      "AES2T5JPFP"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Y4WACJKC3S/",
  "links": [
    {
      "field": "defines",
      "target": "RK2N3QP4FQ"
    },
    {
      "field": "uses",
      "target": "FP5GVCN8GV"
    },
    {
      "field": "uses",
      "target": "CG9JEATW3Q"
    },
    {
      "field": "uses",
      "target": "AES2T5JPFP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y4WACJKC3S.md",
  "msc": [
    "46-XX"
  ],
  "name": "Attainable null space",
  "prov": {
    "log": "https://github.com/tamnd/math2/blob/main/prov/Y4W/Y4WACJKC3S.jsonl",
    "quarantine": null,
    "unchecked": null
  },
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y4W/Y4WACJKC3S.md",
  "status": null,
  "tag": "Y4WACJKC3S",
  "uri": "m2://Y4WACJKC3S",
  "url": "https://math2.tamnd.com/wiki/attainable-null-space-definition/",
  "writing": {
    "tag": "Y4WACJKC3S",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}