{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:49df86f1-e435-4072-ad8f-2cf200b89a68",
  "labels": {
    "zh-CN": "自伴算符的谱定理",
    "en": "Spectral theorem for self-adjoint operators"
  },
  "alternative_labels": [],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "dc68f142871ec12b8e723744948330fe678b92a1f853b5d8185f7d59a7646f75",
  "occurrences": [
    {
      "concept_id": "urn:uuid:49df86f1-e435-4072-ad8f-2cf200b89a68",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "zh-CN",
      "route": "zh/chapter-13/06-unbounded-operators",
      "anchor": "first-0c7c88a6b2004b097883-zh",
      "chapter": "13",
      "section": "13.6",
      "offset": 21458,
      "range": [
        28060,
        28676
      ],
      "source_revision": "d45e53fc27c5bf1d2c2121c20ab0e61aeb57067de09a90fd6173850a6fa3d4f2",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "theorem",
      "role_basis": {
        "kind": "registered-source-environment",
        "registered_position": {
          "route": "zh/chapter-13/06-unbounded-operators",
          "lang": "zh-CN",
          "section": "13.6",
          "signal": "named-theorem",
          "label": "自伴算符的谱定理",
          "english": "Spectral theorem for self-adjoint operators",
          "offset": 18537
        }
      },
      "label": "> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-25\">定理 13.25</a> · 自伴算符的谱定理 (Spectral theorem for self-adjoint operators)",
      "context": {
        "range": [
          28060,
          28676
        ],
        "quote": "\n\n> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-25\">定理 13.25</a> · 自伴算符的谱定理 (Spectral theorem for self-adjoint operators)\n>\n> 令 $A$ 是希尔伯特空间 $\\mathcal{H}$ 上的自伴算子。存在递增投影算子 $E_{\\lambda}(\\lambda \\in \\mathbb{R})$，其中对于 $\\lambda \\leq \\lambda^{\\prime}$，有$E_{\\lambda} \\leq P_{\\lambda}$，使得\n>\n> $$\n> E_{-\\infty}=0 \\quad \\text {且} \\quad E_{\\infty}=I\n> $$\n>\n> 使得\n>\n> $$\n> A=\\int_{-\\infty}^{\\infty} \\lambda \\mathrm{d} E_{\\lambda},\n> $$\n>\n> 其中积分被解释为 勒贝格–斯蒂尔切斯 积分\n>\n> $$\n> \\langle u \\mid A u\\rangle=\\int_{-\\infty}^{\\infty} \\lambda \\mathrm{d}\\left\\langle u \\mid E_{\\lambda} u\\right\\rangle\n> $$\n>\n> 对全体$u \\in D_{A}$都是适用的\n",
        "quote_sha256": "dea8ecffce7c573b77fcd8e05c4d3cce0451a3ceaced880a85b239000890bf28",
        "environment": "theorem",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "en/chapter-13/06-unbounded-operators",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-86ab8f63f1aec8e2502f6b77"
    },
    {
      "concept_id": "urn:uuid:49df86f1-e435-4072-ad8f-2cf200b89a68",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "en",
      "route": "en/chapter-13/06-unbounded-operators",
      "anchor": "first-0c7c88a6b2004b097883-en",
      "chapter": "13",
      "section": "13.6",
      "offset": 28745,
      "range": [
        34840,
        35654
      ],
      "source_revision": "c9f6aaaf4583a87d844433adf5551a93a8faeb812cbe88bab00cc9cdaa02d709",
      "source_category": "original_transcription",
      "review_status": "unreviewed",
      "role": "theorem",
      "role_basis": {
        "kind": "registered-source-environment",
        "registered_position": {
          "route": "en/chapter-13/06-unbounded-operators",
          "lang": "en",
          "section": "13.6",
          "signal": "named-theorem",
          "label": "Spectral theorem for self-adjoint operators",
          "english": "Spectral theorem for self-adjoint operators",
          "offset": 25264
        }
      },
      "label": "> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-25\">Theorem 13.25</a> · Spectral theorem for self-adjoint operators",
      "context": {
        "range": [
          34840,
          35654
        ],
        "quote": "\n\n> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-25\">Theorem 13.25</a> · Spectral theorem for self-adjoint operators\n>\n> Let $A$ be a self-adjoint operator on a Hilbert space $\\mathcal H$. There exists an increasingfamily ofprojection operators $E _ { \\lambda } \\left( \\lambda \\in \\mathbb { R } \\right)$ , with $E _ { \\lambda } \\le P _ { \\lambda ^ { \\prime } } \\text{for} \\lambda \\le \\lambda ^ { \\prime }$ , such that\n>\n> $$\n> E _ {- \\infty} = 0 \\text{and} E _ {\\infty} = I\n> $$\n>\n> such that\n>\n> $$\n> A = \\int_ {- \\infty} ^ {\\infty} \\lambda \\mathrm{d} E _ {\\lambda},\n> $$\n>\n> where the integral is interpreted as the Lebesgue–Stieltjes integra\n>\n> $$\n> \\langle u \\mid A u \\rangle = \\int_ {- \\infty} ^ {\\infty} \\lambda d \\langle u \\mid E _ {\\lambda} u \\rangle\n> $$\n>\n> valid for all $u \\in D _ { A }$\n>",
        "quote_sha256": "c4b23523d51d881c6ef934c9e40bf65202c63d001325e7c2f949fb49425b59e6",
        "environment": "theorem",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "zh/chapter-13/06-unbounded-operators",
          "lang": "zh-CN",
          "source_category": "translation",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-774425528054e7679872f40b"
    }
  ],
  "registered_positions": [
    {
      "route": "en/chapter-13/06-unbounded-operators",
      "lang": "en",
      "section": "13.6",
      "signal": "named-theorem",
      "label": "Spectral theorem for self-adjoint operators",
      "english": "Spectral theorem for self-adjoint operators",
      "offset": 25264
    },
    {
      "route": "zh/chapter-13/06-unbounded-operators",
      "lang": "zh-CN",
      "section": "13.6",
      "signal": "named-theorem",
      "label": "自伴算符的谱定理",
      "english": "Spectral theorem for self-adjoint operators",
      "offset": 18537
    }
  ],
  "statements": [],
  "resolved_references": [
    {
      "raw": "href=\"#theorem-13-25\"",
      "href": "#theorem-13-25",
      "range": [
        28103,
        28124
      ],
      "relation": "references",
      "target_route": "zh/chapter-13/06-unbounded-operators",
      "target_anchor": "theorem-13-25",
      "resolved": true,
      "reason": null,
      "source_revision": "d45e53fc27c5bf1d2c2121c20ab0e61aeb57067de09a90fd6173850a6fa3d4f2",
      "source_route": "zh/chapter-13/06-unbounded-operators",
      "occurrence_id": "occ-86ab8f63f1aec8e2502f6b77"
    },
    {
      "raw": "href=\"#theorem-13-25\"",
      "href": "#theorem-13-25",
      "range": [
        34883,
        34904
      ],
      "relation": "references",
      "target_route": "en/chapter-13/06-unbounded-operators",
      "target_anchor": "theorem-13-25",
      "resolved": true,
      "reason": null,
      "source_revision": "c9f6aaaf4583a87d844433adf5551a93a8faeb812cbe88bab00cc9cdaa02d709",
      "source_route": "en/chapter-13/06-unbounded-operators",
      "occurrence_id": "occ-774425528054e7679872f40b"
    }
  ],
  "reading_order": [
    "occ-86ab8f63f1aec8e2502f6b77",
    "occ-774425528054e7679872f40b"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 2,
    "located_occurrences": 2,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "ddf0f9227c9093e76b4a1a5d1b41fe5a533c7e039a41c791d87dff005ad484b3",
  "optional_enrichments": []
}