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  "body": "\n## Summary\n\nFor a continuous [[seminorm|FP5GVCN8GV]] $q$ on the [[Schwartz space|CG9JEATW3Q]] $\\mathcal{S}$, the **attainable null space** $\\mathcal{N}_q$ is a set of [[tempered distributions|AES2T5JPFP]].\n\nA distribution $p$ in $\\mathcal{S}'$ lies in $\\mathcal{N}_q$ when some sequence $(\\chi_k)$ in $\\mathcal{S}$ converges to $p$ in $\\mathcal{S}'$ and has $q(\\chi_k)\\to 0$.\n\nBecause $q$ is a seminorm, $\\mathcal{N}_q$ is a [[linear subspace|00YC1D652K]] of $\\mathcal{S}'$: sums and scalar multiples of such sequences are again such sequences.\n\nSo $\\mathcal{N}_q$ collects the distributions that test functions of vanishing $q$-size can approximate.\n\nFor a positive-semi-definite operator $\\mathrm{B}$ from $\\mathcal{S}$ to $\\mathcal{S}'$, the set $\\mathcal{N}_{\\mathrm{B}}$ is the case where $q$ is the seminorm induced by $\\mathrm{B}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "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": [
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    "kind": "concept",
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    "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)"
      }
    ],
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      "00YC1D652K"
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    "tag": "RK2N3QP4FQ",
    "uses": [
      "FP5GVCN8GV",
      "CG9JEATW3Q",
      "AES2T5JPFP"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/RK2N3QP4FQ/",
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      "target": "AES2T5JPFP"
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  "markdown": "https://math2.tamnd.com/node/RK2N3QP4FQ.md",
  "msc": [
    "46-XX"
  ],
  "name": "Attainable null space",
  "prov": {
    "log": "https://github.com/tamnd/math2/blob/main/prov/RK2/RK2N3QP4FQ.jsonl",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RK2/RK2N3QP4FQ.md",
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  "tag": "RK2N3QP4FQ",
  "uri": "m2://RK2N3QP4FQ",
  "url": "https://math2.tamnd.com/wiki/attainable-null-space/",
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    "tag": "RK2N3QP4FQ",
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        "what": "edge instance_of in",
        "note": "at least two examples",
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