{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-banach-space",
  "semantic_family_id": "sf-real-banach-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實Banach空間"
  },
  "summary": {
    "zh-Hant": "在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。"
  },
  "statement": {
    "zh-Hant": "實Banach空間必須把同一載體、實向量運算、範數、由該範數誘導的度量及完備性綁成一體。本輪使用有限維R^n配命名l1或linfinity範數的精確定理證書，不以序列抽樣證完備。"
  },
  "formula": {
    "tex": "B(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f26780f0baf666ea25e0c82fb0dacd0beb8058d9e9c6d2cb7ca7b9523a8f443b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "空間為明示有限維R^n，範數限命名l1或linfinity，metric必須由同一norm誘導。"
    },
    {
      "id": "a2",
      "zh-Hant": "完備證書使用有限維實賦範空間的命名定理；有限條Cauchy序列樣本不足。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "real_banach_space",
      "role_zh": "實Banach空間",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_normed_vector_space",
      "role_zh": "同一實賦範空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-normed-vector-space",
      "title_zh": "實賦範向量空間",
      "reason_zh": "Banach先是一個實賦範空間。"
    },
    {
      "id": "mko-norm-induced-metric",
      "title_zh": "範數誘導度量",
      "reason_zh": "完備性使用同一norm的metric。"
    },
    {
      "id": "mko-complete-metric-space",
      "title_zh": "完備度量空間",
      "reason_zh": "所有Cauchy序列須在空間內收斂。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "載體與運算須為同一實向量空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "norm與metric不可來自不同空間或不同公式。",
      "有限維R^n的命名範數提供完備性定理證書。",
      "Banach不自動具有內積或Hilbert結構。"
    ],
    "common_misconception_zh": "測過幾條Cauchy序列，或只知道metric完備，就宣稱任意normed space是Banach。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "命名證書只涵蓋明示有限維R^n與同一norm；不推出Hilbert、緊緻或任意無限維結論。",
    "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": "與內積誘導範數共同支撐實Hilbert空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-banach-space.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-10-28 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"
  }
}