第 15281 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":[[7,2],[2,5]],"x":[1,-2],"y":[2,-7]}。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":[[7,2],[2,5]],"x":[1,-2],"y":[2,-7]}。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(7, 5)

Hilbert Gram diagonal product = 35

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

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(35, 4)

Hilbert Gram determinant = 31

Hilbert completing-square slope := div(2, 7)

Hilbert completing-square slope = 2/7

Hilbert completing-square positive residual := div(31, 7)

Hilbert completing-square positive residual = 31/7

complete real carrier R^2;Gram matrix=[[7, 2], [2, 5]];a=7>0;det=31>0;dimension declaration=2

x^T M x=7*(x0+(2/7)*x1)^2+(31/7)*x1^2;zero only at x=0

Hilbert cross inner product coefficient(0,0) times left coordinate := mul(7, 1)

Hilbert cross inner product coefficient(0,0) times left coordinate = 7

Hilbert cross inner product term(0,0) := mul(7, 2)

Hilbert cross inner product term(0,0) = 14

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, -7)

Hilbert cross inner product term(0,1) = -14

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

Hilbert cross inner product coefficient(1,0) times left coordinate = -4

Hilbert cross inner product term(1,0) := mul(-4, 2)

Hilbert cross inner product term(1,0) = -8

Hilbert cross inner product coefficient(1,1) times left coordinate := mul(5, -2)

Hilbert cross inner product coefficient(1,1) times left coordinate = -10

Hilbert cross inner product term(1,1) := mul(-10, -7)

Hilbert cross inner product term(1,1) = 70

Hilbert cross inner product complete bilinear sum := sum(14, -14, -8, 70)

Hilbert cross inner product complete bilinear sum = 62

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

Hilbert vector sum coordinate0 = 3

Hilbert vector sum coordinate1 := sum(-2, -7)

Hilbert vector sum coordinate1 = -9

Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(7, 3)

Hilbert norm square of sum coefficient(0,0) times left coordinate = 21

Hilbert norm square of sum term(0,0) := mul(21, 3)

Hilbert norm square of sum term(0,0) = 63

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, -9)

Hilbert norm square of sum term(0,1) = -54

Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(2, -9)

Hilbert norm square of sum coefficient(1,0) times left coordinate = -18

Hilbert norm square of sum term(1,0) := mul(-18, 3)

Hilbert norm square of sum term(1,0) = -54

Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(5, -9)

Hilbert norm square of sum coefficient(1,1) times left coordinate = -45

Hilbert norm square of sum term(1,1) := mul(-45, -9)

Hilbert norm square of sum term(1,1) = 405

Hilbert norm square of sum complete bilinear sum := sum(63, -54, -54, 405)

Hilbert norm square of sum complete bilinear sum = 360

Hilbert first norm square coefficient(0,0) times left coordinate := mul(7, 1)

Hilbert first norm square coefficient(0,0) times left coordinate = 7

Hilbert first norm square term(0,0) := mul(7, 1)

Hilbert first norm square term(0,0) = 7

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, -2)

Hilbert first norm square term(0,1) = -4

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

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

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

Hilbert first norm square term(1,0) = -4

Hilbert first norm square coefficient(1,1) times left coordinate := mul(5, -2)

Hilbert first norm square coefficient(1,1) times left coordinate = -10

Hilbert first norm square term(1,1) := mul(-10, -2)

Hilbert first norm square term(1,1) = 20

Hilbert first norm square complete bilinear sum := sum(7, -4, -4, 20)

Hilbert first norm square complete bilinear sum = 19

Hilbert second norm square coefficient(0,0) times left coordinate := mul(7, 2)

Hilbert second norm square coefficient(0,0) times left coordinate = 14

Hilbert second norm square term(0,0) := mul(14, 2)

Hilbert second norm square term(0,0) = 28

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, -7)

Hilbert second norm square term(0,1) = -28

Hilbert second norm square coefficient(1,0) times left coordinate := mul(2, -7)

Hilbert second norm square coefficient(1,0) times left coordinate = -14

Hilbert second norm square term(1,0) := mul(-14, 2)

Hilbert second norm square term(1,0) = -28

Hilbert second norm square coefficient(1,1) times left coordinate := mul(5, -7)

Hilbert second norm square coefficient(1,1) times left coordinate = -35

Hilbert second norm square term(1,1) := mul(-35, -7)

Hilbert second norm square term(1,1) = 245

Hilbert second norm square complete bilinear sum := sum(28, -28, -28, 245)

Hilbert second norm square complete bilinear sum = 217

Hilbert sum of separate norm squares := sum(19, 217)

Hilbert sum of separate norm squares = 236

Hilbert cross term twice inner product := mul(2, 62)

Hilbert cross term twice inner product = 124

Hilbert expanded square including cross term := sum(236, 124)

Hilbert expanded square including cross term = 360

same inner product=62;norm square of sum=360;separate squares sum=236;twice cross term=124

bilinear expansion=360;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, -2], [2, -7], 62, 360, 236, 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。