{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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-real-normed-vector-space"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-real-normed-vector-space-01","title":{"zh-Hant":"實賦範向量空間：ℓ¹ 與 ℓ^∞ 公式的全稱範數證書，以及兩個非範數公式","en":"Real normed vector space: universal norm certificates for the ℓ¹ and ℓ^∞ formulas, and two formulas that are not norms"},"summary":{"zh-Hant":"在 ℝⁿ（n≥1 固定）上，對所有向量 u、v 與所有實數 a，證明 ‖v‖₁=Σ|v_i| 與 ‖v‖_∞=max|v_i| 都滿足範數的三條公理：非負且 ‖v‖=0 ⇔ v=0、‖av‖=|a|‖v‖、‖u+v‖≤‖u‖+‖v‖。再在 ℝ³ 中對整數座標向量 u=(3,−4,1)、v=(−1,2,5) 計算兩種範數與三角不等式、齊次性的實例；最後說明 φ(v)=|v₁| 與 ψ(v)=(‖v‖₁)² 不是範數。","en":"On ℝⁿ (n≥1 fixed), for all vectors u, v and all reals a, prove that ‖v‖₁=Σ|v_i| and ‖v‖_∞=max|v_i| both satisfy the three norm axioms: nonnegativity with ‖v‖=0 ⇔ v=0, ‖av‖=|a|‖v‖, and ‖u+v‖≤‖u‖+‖v‖. Then, in ℝ³, compute both norms and instances of the triangle inequality and homogeneity for the integer-coordinate vectors u=(3,−4,1) and v=(−1,2,5); finally show that φ(v)=|v₁| and ψ(v)=(‖v‖₁)² are not norms."},"mko_ids":["mko-real-normed-vector-space"]}],"title":"概念推導導讀"}