第 13458 題：實Banach空間

原題與已知條件

載體norm誘導metric與完備證書必須描述同一空間。操作=banach-evidence-mode-audit；資料={"dimension":8,"norm_name":"linfinity","evidence_mode":"finite_dimensional_theorem"}。請依 [dimension,norm_name,evidence_mode,sufficient,boundary] 欄序回答。

• G001 — 載體norm誘導metric與完備證書必須描述同一空間。操作=banach-evidence-mode-audit；資料={"dimension":8,"norm_name":"linfinity","evidence_mode":"finite_dimensional_theorem"}。請依 [dimension,norm_name,evidence_mode,sufficient,boundary] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-real-banach-space

逐步推導

same real carrier R^8;standard coordinate vector operations;named norm=linfinity

metric is d(x,y)=this_same_norm(x-y);norm axioms supplied by named formula

arbitrary Cauchy sequence x_k in R^8 under linfinity;every coordinate obeys abs((x_k-x_m)_i)<=norm(x_k-x_m)

each real coordinate sequence is Cauchy;real-number completeness supplies a real limit L_i

coordinate limits L_i form L in R^8;epsilon>0;each coordinate error eventually <epsilon

choose N=max of the 8 coordinate cutoffs;maximum of finitely many coordinate errors < epsilon;norm(x_k-L)->0

every Cauchy sequence in this named real R^8 normed space has a same-space norm limit

Banach certificate=是;no Hilbert or arbitrary infinite-dimensional conclusion

named real space R^8;norm=linfinity;evidence=finite_dimensional_theorem

sufficient evidence=是;sample insufficiency is not incompleteness

逐步反向核對

• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(8, linfinity, finite_dimensional_theorem, 是, universal_named_certificate)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。