{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-normed-vector-space",
  "semantic_family_id": "sf-real-normed-vector-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實賦範向量空間"
  },
  "summary": {
    "zh-Hant": "實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。"
  },
  "statement": {
    "zh-Hant": "範數把向量送到非負實數，且norm(v)=0當且僅當v=0、norm(a v)=|a| norm(v)、norm(u+v)不超過norm(u)+norm(v)。本輪只使用R^n上的l1與linfinity命名公式。"
  },
  "formula": {
    "tex": "N(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>N</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "15078602ffe4bd22e6828ec651f79e8bf16f7d6c41eec1ae231d17369d172fb8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫用固定有限維整數座標向量。"
    },
    {
      "id": "a2",
      "zh-Hant": "l1與linfinity資格來自公式的全稱代數證書；有限抽樣不冒充一般範數證明。"
    }
  ],
  "symbols": [
    {
      "token": "N",
      "role": "norm_record",
      "role_zh": "範數紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_vector_space",
      "role_zh": "實向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "範數定義在實向量空間。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "純量齊次使用實數絕對值。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "非負與三角不等式依實數順序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "l1是座標絕對值總和。",
      "linfinity是座標絕對值最大值。",
      "範數差可誘導兩點距離，但誘導度量另立節點。"
    ],
    "common_misconception_zh": "以有限點通過非負與三角測試就宣稱任意公式是範數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只接受命名精確公式；不從抽樣建立範數公理。",
    "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": "加入範數誘導度量、Banach與有界算子。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-normed-vector-space.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-26 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}