{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-bounded-linear-operator"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-bounded-linear-operator-01","title":{"zh-Hant":"有界線性算子：矩陣在 ℓ^∞ 與 ℓ¹ 下的精確誘導算子範數","en":"Bounded linear operator: exact induced operator norms of a matrix under ℓ^∞ and ℓ¹"},"summary":{"zh-Hant":"設 A 為第一行 (1,−2)、第二行 (3,4) 的整數矩陣，T(x)=Ax:ℝ²→ℝ²。（一）在兩端都用 ℓ^∞ 範數時，證明對所有 x 有 ‖Ax‖_∞≤7‖x‖_∞，且沒有更小的常數可用，所以 ‖T‖_{∞→∞}=7（最大行絕對值和）。（二）兩端都用 ℓ¹ 範數時，證明 ‖T‖_{1→1}=6（最大列絕對值和）。（三）說明非最佳常數、有限輸入與值域無界這三個陷阱。","en":"Let A be the integer matrix with first row (1,−2) and second row (3,4), and T(x)=Ax:ℝ²→ℝ². (1) With the ℓ^∞ norm at both ends, prove that ‖Ax‖_∞≤7‖x‖_∞ for all x and that no smaller constant works, so ‖T‖_{∞→∞}=7 (the maximum absolute row sum). (2) With the ℓ¹ norm at both ends, prove ‖T‖_{1→1}=6 (the maximum absolute column sum). (3) Explain three traps: non-optimal constants, finitely many inputs, and unbounded range."},"mko_ids":["mko-bounded-linear-operator"]}],"title":"Concept derivation guides"}