第 15299 題：實Hilbert空間

原題與已知條件

inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={"matrix":[[25,2],[2,5]],"x":[1,1],"y":[2,-25]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。

• G001 — inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={"matrix":[[25,2],[2,5]],"x":[1,1],"y":[2,-25]}。請依 [x,y,inner,lhs,rhs,result] 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-hilbert-space

逐步推導

Hilbert Gram diagonal product := mul(25, 5)

Hilbert Gram diagonal product = 125

Hilbert Gram off-diagonal product := mul(2, 2)

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(125, 4)

Hilbert Gram determinant = 121

Hilbert completing-square slope := div(2, 25)

Hilbert completing-square slope = 2/25

Hilbert completing-square positive residual := div(121, 25)

Hilbert completing-square positive residual = 121/25

complete real carrier R^2;Gram matrix=[[25, 2], [2, 5]];a=25>0;det=121>0;dimension declaration=2

x^T M x=25*(x0+(2/25)*x1)^2+(121/25)*x1^2;zero only at x=0

Hilbert cross inner product coefficient(0,0) times left coordinate := mul(25, 1)

Hilbert cross inner product coefficient(0,0) times left coordinate = 25

Hilbert cross inner product term(0,0) := mul(25, 2)

Hilbert cross inner product term(0,0) = 50

Hilbert cross inner product coefficient(0,1) times left coordinate := mul(2, 1)

Hilbert cross inner product coefficient(0,1) times left coordinate = 2

Hilbert cross inner product term(0,1) := mul(2, -25)

Hilbert cross inner product term(0,1) = -50

Hilbert cross inner product coefficient(1,0) times left coordinate := mul(2, 1)

Hilbert cross inner product coefficient(1,0) times left coordinate = 2

Hilbert cross inner product term(1,0) := mul(2, 2)

Hilbert cross inner product term(1,0) = 4

Hilbert cross inner product coefficient(1,1) times left coordinate := mul(5, 1)

Hilbert cross inner product coefficient(1,1) times left coordinate = 5

Hilbert cross inner product term(1,1) := mul(5, -25)

Hilbert cross inner product term(1,1) = -125

Hilbert cross inner product complete bilinear sum := sum(50, -50, 4, -125)

Hilbert cross inner product complete bilinear sum = -121

Hilbert vector sum coordinate0 := sum(1, 2)

Hilbert vector sum coordinate0 = 3

Hilbert vector sum coordinate1 := sum(1, -25)

Hilbert vector sum coordinate1 = -24

Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(25, 3)

Hilbert norm square of sum coefficient(0,0) times left coordinate = 75

Hilbert norm square of sum term(0,0) := mul(75, 3)

Hilbert norm square of sum term(0,0) = 225

Hilbert norm square of sum coefficient(0,1) times left coordinate := mul(2, 3)

Hilbert norm square of sum coefficient(0,1) times left coordinate = 6

Hilbert norm square of sum term(0,1) := mul(6, -24)

Hilbert norm square of sum term(0,1) = -144

Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(2, -24)

Hilbert norm square of sum coefficient(1,0) times left coordinate = -48

Hilbert norm square of sum term(1,0) := mul(-48, 3)

Hilbert norm square of sum term(1,0) = -144

Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(5, -24)

Hilbert norm square of sum coefficient(1,1) times left coordinate = -120

Hilbert norm square of sum term(1,1) := mul(-120, -24)

Hilbert norm square of sum term(1,1) = 2880

Hilbert norm square of sum complete bilinear sum := sum(225, -144, -144, 2880)

Hilbert norm square of sum complete bilinear sum = 2817

Hilbert first norm square coefficient(0,0) times left coordinate := mul(25, 1)

Hilbert first norm square coefficient(0,0) times left coordinate = 25

Hilbert first norm square term(0,0) := mul(25, 1)

Hilbert first norm square term(0,0) = 25

Hilbert first norm square coefficient(0,1) times left coordinate := mul(2, 1)

Hilbert first norm square coefficient(0,1) times left coordinate = 2

Hilbert first norm square term(0,1) := mul(2, 1)

Hilbert first norm square term(0,1) = 2

Hilbert first norm square coefficient(1,0) times left coordinate := mul(2, 1)

Hilbert first norm square coefficient(1,0) times left coordinate = 2

Hilbert first norm square term(1,0) := mul(2, 1)

Hilbert first norm square term(1,0) = 2

Hilbert first norm square coefficient(1,1) times left coordinate := mul(5, 1)

Hilbert first norm square coefficient(1,1) times left coordinate = 5

Hilbert first norm square term(1,1) := mul(5, 1)

Hilbert first norm square term(1,1) = 5

Hilbert first norm square complete bilinear sum := sum(25, 2, 2, 5)

Hilbert first norm square complete bilinear sum = 34

Hilbert second norm square coefficient(0,0) times left coordinate := mul(25, 2)

Hilbert second norm square coefficient(0,0) times left coordinate = 50

Hilbert second norm square term(0,0) := mul(50, 2)

Hilbert second norm square term(0,0) = 100

Hilbert second norm square coefficient(0,1) times left coordinate := mul(2, 2)

Hilbert second norm square coefficient(0,1) times left coordinate = 4

Hilbert second norm square term(0,1) := mul(4, -25)

Hilbert second norm square term(0,1) = -100

Hilbert second norm square coefficient(1,0) times left coordinate := mul(2, -25)

Hilbert second norm square coefficient(1,0) times left coordinate = -50

Hilbert second norm square term(1,0) := mul(-50, 2)

Hilbert second norm square term(1,0) = -100

Hilbert second norm square coefficient(1,1) times left coordinate := mul(5, -25)

Hilbert second norm square coefficient(1,1) times left coordinate = -125

Hilbert second norm square term(1,1) := mul(-125, -25)

Hilbert second norm square term(1,1) = 3125

Hilbert second norm square complete bilinear sum := sum(100, -100, -100, 3125)

Hilbert second norm square complete bilinear sum = 3025

Hilbert sum of separate norm squares := sum(34, 3025)

Hilbert sum of separate norm squares = 3059

Hilbert cross term twice inner product := mul(2, -121)

Hilbert cross term twice inner product = -242

Hilbert expanded square including cross term := sum(3059, -242)

Hilbert expanded square including cross term = 2817

same inner product=-121;norm square of sum=2817;separate squares sum=3059;twice cross term=-242

bilinear expansion=2817;orthogonality premise=否;Pythagorean conclusion status=orthogonality_premise_false

逐步反向核對

• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 Hilbert expanded square including cross term，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Hilbert cross term twice inner product，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Hilbert sum of separate norm squares，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square complete bilinear sum，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square term(1,1)，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square coefficient(1,1) times left coordinate，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square term(1,0)，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square coefficient(1,0) times left coordinate，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square term(0,1)，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square coefficient(0,1) times left coordinate，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square term(0,0)，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Hilbert second norm square coefficient(0,0) times left coordinate，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square complete bilinear sum，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square term(1,1)，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square coefficient(1,1) times left coordinate，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square term(1,0)，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square coefficient(1,0) times left coordinate，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square term(0,1)，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square coefficient(0,1) times left coordinate，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square term(0,0)，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Hilbert first norm square coefficient(0,0) times left coordinate，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum complete bilinear sum，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum term(1,1)，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum coefficient(1,1) times left coordinate，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum term(1,0)，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum coefficient(1,0) times left coordinate，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum term(0,1)，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum coefficient(0,1) times left coordinate，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum term(0,0)，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Hilbert norm square of sum coefficient(0,0) times left coordinate，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Hilbert vector sum coordinate1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Hilbert vector sum coordinate0，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product complete bilinear sum，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product term(1,1)，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product coefficient(1,1) times left coordinate，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product term(1,0)，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product coefficient(1,0) times left coordinate，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product term(0,1)，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product coefficient(0,1) times left coordinate，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product term(0,0)，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Hilbert cross inner product coefficient(0,0) times left coordinate，不以小數近似替代。
• S006 · 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Hilbert completing-square positive residual，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Hilbert completing-square slope，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Hilbert Gram determinant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Hilbert Gram off-diagonal product，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 Hilbert Gram diagonal product，不以小數近似替代。

結論

([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false)

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。