{"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-norm-induced-metric"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-norm-induced-metric-01","title":{"zh-Hant":"範數誘導度量：由範數公理證明 d(x,y)=‖x−y‖ 是度量，並在 ℝ³ 上以 l1 與 l∞ 精確計算","en":"Norm-induced metric: proving from the norm axioms that d(x,y)=‖x−y‖ is a metric, with exact l1 and l∞ computations in ℝ³"},"summary":{"zh-Hant":"設 (V,‖·‖) 是實賦範向量空間。證明 d(x,y)=‖x−y‖ 滿足非負、分離、對稱與三角不等式；證明 ℝⁿ 上的 l1 與 l∞ 公式都是範數，所以都誘導度量；再在 ℝ³ 上對 x=(1,−2,3)、y=(4,0,−1)、z=(2,5,1) 精確計算 d1 與 d∞。","en":"Let (V,‖·‖) be a real normed vector space. Prove that d(x,y)=‖x−y‖ is nonnegative, separates points, is symmetric and satisfies the triangle inequality; prove that the l1 and l∞ formulas on ℝⁿ are norms, so both induce metrics; then compute d1 and d∞ exactly in ℝ³ for x=(1,−2,3), y=(4,0,−1), z=(2,5,1)."},"mko_ids":["mko-norm-induced-metric"]}],"title":"Concept derivation guides"}