{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7",
  "labels": {
    "zh-CN": "对偶向量空间",
    "en": "Dual vector space"
  },
  "alternative_labels": [
    {
      "lang": "en",
      "value": "dual vector space"
    }
  ],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "e26cdabe021537717a7bb7a578935a9d0380b80168fdae5211e940e167d3942d",
  "occurrences": [
    {
      "concept_id": "urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
      "anchor": "first-9224a02242ce94330051-en",
      "chapter": "4",
      "section": "4.2",
      "offset": 502,
      "range": [
        7780,
        8715
      ],
      "source_revision": "b22c3ff4ebb657bf1e6ea12bb32e608b2c606878c532aeaa3bee39833ba66c12",
      "source_category": "original_transcription",
      "review_status": "ai-transcription-compared",
      "role": "definition",
      "role_basis": {
        "kind": "registered-source-environment",
        "registered_position": {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
          "lang": "en",
          "section": "4.2",
          "signal": "named-definition",
          "label": "Dual vector space",
          "english": "Dual vector space",
          "offset": 432
        }
      },
      "label": "> [!definition] Definition · <span class=\"definition-name\">Dual vector space</span>",
      "context": {
        "range": [
          7780,
          8715
        ],
        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Dual vector space</span>\n> <span id=\"source-block-024\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u id=\"concept-4-dual-vector-space\" class=\"concept-term\" data-concept-id=\"concept-4-dual-vector-space\" data-concept-identity=\"urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\" data-concept-route=\"/concept-identities#concept-07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\">Dual vector space</u>). </span>For $V$ a vector space over a field $k$, the dual vector space $V ^ { * }$ is the vector space of all linear maps $V \\to k$, i.e.,\n>\n> <span id=\"source-block-025\" class=\"block-anchor\"></span>\n>\n> $$\n> V ^ {*} = \\{l: V \\rightarrow k \\text {   such   that   } l (\\alpha v + \\beta w) = \\alpha l (v) + \\beta l (w) \\}\n> $$\n>\n> <span id=\"source-block-026\" class=\"block-anchor\"></span>\n>\n> for $\\alpha , \\beta \\in k , \\ v , w \\in V$.\n",
        "quote_sha256": "152acd7aa50072f8a276e8469ac740e13e38bfa73682342105d3afe8714bddab",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
          "lang": "zh-CN",
          "source_category": "translation",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-738ff8b42b8c3db3e3aa2d33"
    },
    {
      "concept_id": "urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
      "anchor": "concept-4-dual-vector-space",
      "chapter": "4",
      "section": "4.2",
      "offset": 720,
      "range": [
        7998,
        8278
      ],
      "source_revision": "b22c3ff4ebb657bf1e6ea12bb32e608b2c606878c532aeaa3bee39833ba66c12",
      "source_category": "original_transcription",
      "review_status": "ai-transcription-compared",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "Dual vector space",
      "context": {
        "range": [
          7780,
          8715
        ],
        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Dual vector space</span>\n> <span id=\"source-block-024\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u id=\"concept-4-dual-vector-space\" class=\"concept-term\" data-concept-id=\"concept-4-dual-vector-space\" data-concept-identity=\"urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\" data-concept-route=\"/concept-identities#concept-07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\">Dual vector space</u>). </span>For $V$ a vector space over a field $k$, the dual vector space $V ^ { * }$ is the vector space of all linear maps $V \\to k$, i.e.,\n>\n> <span id=\"source-block-025\" class=\"block-anchor\"></span>\n>\n> $$\n> V ^ {*} = \\{l: V \\rightarrow k \\text {   such   that   } l (\\alpha v + \\beta w) = \\alpha l (v) + \\beta l (w) \\}\n> $$\n>\n> <span id=\"source-block-026\" class=\"block-anchor\"></span>\n>\n> for $\\alpha , \\beta \\in k , \\ v , w \\in V$.\n",
        "quote_sha256": "152acd7aa50072f8a276e8469ac740e13e38bfa73682342105d3afe8714bddab",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
          "lang": "zh-CN",
          "source_category": "translation",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-5abd38214ad6696e790db376"
    },
    {
      "concept_id": "urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
      "anchor": "first-9224a02242ce94330051-zh",
      "chapter": "4",
      "section": "4.2",
      "offset": 1088,
      "range": [
        9110,
        10466
      ],
      "source_revision": "938d19d061a8b69c758beeb064276f3de2f0639a77ac963f124b6481149d29e7",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "definition",
      "role_basis": {
        "kind": "registered-source-environment",
        "registered_position": {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
          "lang": "zh-CN",
          "section": "4.2",
          "signal": "named-definition",
          "label": "对偶向量空间",
          "english": "dual vector space",
          "offset": 706
        }
      },
      "label": "> [!definition] 定义 · <span class=\"definition-name\">对偶向量空间(dual vector space)</span>",
      "context": {
        "range": [
          9110,
          10466
        ],
        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">对偶向量空间(dual vector space)</span>\n> <span id=\"source-block-024\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（[<u id=\"concept-4-dual-vector-space\" class=\"concept-term\" data-concept-id=\"concept-4-dual-vector-space\" data-concept-role=\"definition\" data-concept-reference=\"concepts/linear-algebra/dual-space\" data-concept-identity=\"urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\" data-concept-route=\"/concept-identities#concept-07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\">对偶向量空间(dual vector space)</u>](../../../../../concepts/linear-algebra/dual-space)）。 </span>对于域 $k$ 上的向量空间 $V$， 对偶向量空间 $V ^ { * }$ 是所有从 $V$ 到 $k$ 的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-4c01999493383d81b50c-zh\">线性映射(linear maps)</u>](../../../../../concepts/linear-algebra/linear-map)构成的向量空间， 即，\n>\n> <span id=\"source-block-025\" class=\"block-anchor\"></span>\n>\n> $$\n> V ^ {*} = \\{l: V \\rightarrow k \\text {   such   that   } l (\\alpha v + \\beta w) = \\alpha l (v) + \\beta l (w) \\}\n> $$\n>\n> <span id=\"source-block-026\" class=\"block-anchor\"></span>\n>\n> 其中 $\\alpha,\\beta\\in k$ 且 $v,w\\in V$。\n",
        "quote_sha256": "feadc197fddc2e45df0fd25e5a615a35e6c975f0b45abb2579d1de1a46eb3fbc",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-d5cf474ea8774426494e97c3"
    },
    {
      "concept_id": "urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
      "anchor": "concept-4-dual-vector-space",
      "chapter": "4",
      "section": "4.2",
      "offset": 1298,
      "range": [
        9320,
        9699
      ],
      "source_revision": "938d19d061a8b69c758beeb064276f3de2f0639a77ac963f124b6481149d29e7",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "对偶向量空间(dual vector space)",
      "context": {
        "range": [
          9110,
          10466
        ],
        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">对偶向量空间(dual vector space)</span>\n> <span id=\"source-block-024\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（[<u id=\"concept-4-dual-vector-space\" class=\"concept-term\" data-concept-id=\"concept-4-dual-vector-space\" data-concept-role=\"definition\" data-concept-reference=\"concepts/linear-algebra/dual-space\" data-concept-identity=\"urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\" data-concept-route=\"/concept-identities#concept-07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\">对偶向量空间(dual vector space)</u>](../../../../../concepts/linear-algebra/dual-space)）。 </span>对于域 $k$ 上的向量空间 $V$， 对偶向量空间 $V ^ { * }$ 是所有从 $V$ 到 $k$ 的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-4c01999493383d81b50c-zh\">线性映射(linear maps)</u>](../../../../../concepts/linear-algebra/linear-map)构成的向量空间， 即，\n>\n> <span id=\"source-block-025\" class=\"block-anchor\"></span>\n>\n> $$\n> V ^ {*} = \\{l: V \\rightarrow k \\text {   such   that   } l (\\alpha v + \\beta w) = \\alpha l (v) + \\beta l (w) \\}\n> $$\n>\n> <span id=\"source-block-026\" class=\"block-anchor\"></span>\n>\n> 其中 $\\alpha,\\beta\\in k$ 且 $v,w\\in V$。\n",
        "quote_sha256": "feadc197fddc2e45df0fd25e5a615a35e6c975f0b45abb2579d1de1a46eb3fbc",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-68cb02e46e3225eeec46a591"
    }
  ],
  "registered_positions": [
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
      "lang": "en",
      "section": "4.2",
      "signal": "named-definition",
      "label": "Dual vector space",
      "english": "Dual vector space",
      "offset": 432
    },
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces",
      "lang": "en",
      "section": "4.2",
      "signal": "term-mark",
      "label": "Dual vector space",
      "english": "Dual vector space",
      "offset": 649
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
      "lang": "zh-CN",
      "section": "4.2",
      "signal": "named-definition",
      "label": "对偶向量空间",
      "english": "dual vector space",
      "offset": 706
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces",
      "lang": "zh-CN",
      "section": "4.2",
      "signal": "term-mark",
      "label": "对偶向量空间",
      "english": "dual vector space",
      "offset": 915
    }
  ],
  "statements": [],
  "resolved_references": [],
  "reading_order": [
    "occ-738ff8b42b8c3db3e3aa2d33",
    "occ-5abd38214ad6696e790db376",
    "occ-d5cf474ea8774426494e97c3",
    "occ-68cb02e46e3225eeec46a591"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 4,
    "located_occurrences": 4,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "c7eb048efaae27879409014c0dcb4c97865d80a946791e7af0ecc4c20029a15d",
  "optional_enrichments": []
}
