{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-hilbert-space",
  "semantic_family_id": "sf-real-hilbert-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實Hilbert空間"
  },
  "summary": {
    "zh-Hant": "在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。"
  },
  "statement": {
    "zh-Hant": "carrier、inner product、induced norm、induced metric與completeness certificate必須描述同一空間。本輪使用有限維R^n標準或正定矩陣內積的命名完備定理證書；抽樣Cauchy序列不足。"
  },
  "formula": {
    "tex": "H(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>H</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "H",
        "glyph": "H"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ee17f8eb0f607c6087b574ba13bdfaca699f719007e5e9845d56bff62485cd9d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "明示有限維R^n與同一正定內積矩陣。"
    },
    {
      "id": "a2",
      "zh-Hant": "完備性使用有限維實內積空間定理；不從樣本推完備。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "real_hilbert_space",
      "role_zh": "實Hilbert空間",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_inner_product_space",
      "role_zh": "同一實內積空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-inner-product-space",
      "title_zh": "實內積空間",
      "reason_zh": "Hilbert先是內積空間。"
    },
    {
      "id": "mko-inner-product-induced-norm",
      "title_zh": "內積誘導範數",
      "reason_zh": "norm來自同一內積。"
    },
    {
      "id": "mko-norm-induced-metric",
      "title_zh": "範數誘導度量",
      "reason_zh": "metric來自同一norm。"
    },
    {
      "id": "mko-complete-metric-space",
      "title_zh": "完備度量空間",
      "reason_zh": "所有Cauchy序列在同一空間收斂。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "五個結構必須同一來源。",
      "有限維R^n提供命名完備證書。",
      "Hilbert不自動推出緊緻、可分或任意維性質。"
    ],
    "common_misconception_zh": "混用不同norm與metric，或用有限序列樣本宣稱完備。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只涵蓋明示有限維證書；不推出compactness、separability或無限維結果。",
    "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": "加入正交投影與Hilbert基底。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-hilbert-space.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-10-30 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"
  }
}