{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"paths","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":{},"path":{"id":"path-pulse-foundations-20261028","title":{"zh-Hant":"結構深度Wave4第二層：交換環、理想、Banach、內積、算子、獨立與連通分量","en":"Structural Depth Wave4 Second Layer: Commutative Rings, Ideals, Banach and Inner-Product Spaces, Operators, Independence and Components"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-normed-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-norm-induced-metric","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complete-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-bilinear-form","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-general-linear-map","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event-independence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-measurable-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph-walk-simple-path","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-commutative-ring","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。","en":"A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist."}},{"id":"mko-finite-two-sided-ideal","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。","en":"A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides."}},{"id":"mko-real-banach-space","position":4,"requirement":"optional","rationale":{"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。","en":"A real normed vector space is a Banach space when it is complete in the metric induced by its own norm."}},{"id":"mko-real-inner-product-space","position":5,"requirement":"optional","rationale":{"zh-Hant":"對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。","en":"A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice."}},{"id":"mko-bounded-linear-operator","position":6,"requirement":"optional","rationale":{"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。","en":"A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms."}},{"id":"mko-random-variable-independence","position":7,"requirement":"optional","rationale":{"zh-Hant":"同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。","en":"Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses."}},{"id":"mko-connected-component","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。","en":"In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path."}}]},"title":"結構深度Wave4第二層：交換環、理想、Banach、內積、算子、獨立與連通分量"}