{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bounded-linear-operator",
  "semantic_family_id": "sf-bounded-linear-operator",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有界線性算子"
  },
  "summary": {
    "zh-Hant": "賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。"
  },
  "statement": {
    "zh-Hant": "T:V到W為有界線性算子，若存在C>=0使每個x皆有norm_W(Tx)<=C norm_V(x)。本輪使用整矩陣在l1或linfinity範數下的精確誘導算子範數證書。"
  },
  "formula": {
    "tex": "O(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>O</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "O",
        "glyph": "O"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7088196ac8eeb47ffbaf05c8468099a504cb2a0fb634ca34a3fab9272a20d84d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "來源與目標為明示有限維實座標空間，兩端norm名稱固定。"
    },
    {
      "id": "a2",
      "zh-Hant": "l1使用最大欄絕對值和、linfinity使用最大列絕對值和；instance通過不替全稱證書。"
    }
  ],
  "symbols": [
    {
      "token": "O",
      "role": "bounded_operator_record",
      "role_zh": "有界算子紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "linear_map",
      "role_zh": "線性映射",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "uniform_bound",
      "role_zh": "統一界常數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-general-linear-map",
      "title_zh": "實向量空間間線性映射",
      "reason_zh": "先有線性映射資格。"
    },
    {
      "id": "mko-real-normed-vector-space",
      "title_zh": "實賦範向量空間",
      "reason_zh": "來源與目標norm須明示。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "統一範數不等式使用非負順序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "同一C必須控制所有輸入。",
      "非最佳C仍可證bounded，但不得冒充operator norm。",
      "矩陣在有限維命名norm下可精確求誘導operator norm。"
    ],
    "common_misconception_zh": "只看有限輸入、把range看似有界當定義，或把任意可用上界稱為operator norm。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核有限維整矩陣與命名norm；不外推任意無限維operator或譜性質。",
    "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": "加入連續線性算子、對偶與譜理論。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bounded-linear-operator.md 整理；2026-10-14 脈衝 R05（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"
  }
}