{
  "$schema": "../../../schemas/mko-v0.5.schema.json",
  "schema_version": "mko-v0.5",
  "id": "mko-dual-vector-space",
  "semantic_family_id": "sf-dual-vector-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "對偶向量空間"
  },
  "summary": {
    "zh-Hant": "實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。"
  },
  "statement": {
    "zh-Hant": "本輪限有限維V=R^n，以row covector給出線性泛函的symbolic全稱證書，並以phi_i(v_j)=delta_ij核dual basis。primal vector與dual functional不混同；沒有額外內積時不宣稱V與V*典範識別。"
  },
  "formula": {
    "tex": "\\varphi\\in V^*",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>φ</mi><mo>∈</mo><msup><mi>V</mi><mo>∗</mo></msup></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "varphi",
        "glyph": "φ"
      },
      "set": {
        "type": "dual_space",
        "space": {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-source13",
      "version": "0.5.0",
      "source_sha256": "ad87a01f0b7ea288eb98bd12aa115fe23f9c7c2747a0fb231b128ed5d05aba70"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "固定有限維實空間與明示有序基底。"
    },
    {
      "id": "a2",
      "zh-Hant": "矩陣、泛函值與換基一律使用精確有理運算。"
    }
  ],
  "symbols": [
    {
      "token": "V^*",
      "role": "dual_vector_space",
      "role_zh": "對偶空間",
      "scope": "representative_formula"
    },
    {
      "token": "\\varphi",
      "role": "linear_functional",
      "role_zh": "線性泛函",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "finite_real_vector_space",
      "role_zh": "有限維實向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-general-linear-map",
      "title_zh": "實向量空間間線性映射",
      "reason_zh": "線性泛函是到R的線性映射。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "來源空間與純量域。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "泛函的值域。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "dual元素是函數而非primal向量。",
      "dual basis要逐對核Kronecker delta。",
      "primal與covector換基律相反；任何識別都需要額外結構。",
      "記號對齊候選：V^* 表示所有實線性泛函 V→R 的對偶空間；代表公式只宣告 φ∈V^*，星號不是未定義的冪指數，也不將向量與泛函典範識別。"
    ],
    "common_misconception_zh": "因座標列長一樣就把向量與泛函視為同一個典範物件。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限維R^n與明示基底；不推無限維連續對偶或自然同構。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入annihilator、double dual與adjoint。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-dual-vector-space.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-05 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}