第 15278 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":[[4,2],[2,5]],"x":[1,0],"y":[2,-4]}。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":[[4,2],[2,5]],"x":[1,0],"y":[2,-4]}。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(4, 5)

Hilbert Gram diagonal product = 20

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

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(20, 4)

Hilbert Gram determinant = 16

Hilbert completing-square slope := div(2, 4)

Hilbert completing-square slope = 1/2

Hilbert completing-square positive residual := div(16, 4)

Hilbert completing-square positive residual = 4

complete real carrier R^2;Gram matrix=[[4, 2], [2, 5]];a=4>0;det=16>0;dimension declaration=2

x^T M x=4*(x0+(1/2)*x1)^2+(4)*x1^2;zero only at x=0

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

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

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

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

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, -4)

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

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

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

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

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

Hilbert cross inner product coefficient(1,1) times left coordinate := mul(5, 0)

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

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

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

Hilbert cross inner product complete bilinear sum := sum(8, -8, 0, 0)

Hilbert cross inner product complete bilinear sum = 0

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

Hilbert vector sum coordinate0 = 3

Hilbert vector sum coordinate1 := sum(0, -4)

Hilbert vector sum coordinate1 = -4

Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(4, 3)

Hilbert norm square of sum coefficient(0,0) times left coordinate = 12

Hilbert norm square of sum term(0,0) := mul(12, 3)

Hilbert norm square of sum term(0,0) = 36

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, -4)

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

Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(2, -4)

Hilbert norm square of sum coefficient(1,0) times left coordinate = -8

Hilbert norm square of sum term(1,0) := mul(-8, 3)

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

Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(5, -4)

Hilbert norm square of sum coefficient(1,1) times left coordinate = -20

Hilbert norm square of sum term(1,1) := mul(-20, -4)

Hilbert norm square of sum term(1,1) = 80

Hilbert norm square of sum complete bilinear sum := sum(36, -24, -24, 80)

Hilbert norm square of sum complete bilinear sum = 68

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

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

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

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

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, 0)

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

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

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

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

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

Hilbert first norm square coefficient(1,1) times left coordinate := mul(5, 0)

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

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

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

Hilbert first norm square complete bilinear sum := sum(4, 0, 0, 0)

Hilbert first norm square complete bilinear sum = 4

Hilbert second norm square coefficient(0,0) times left coordinate := mul(4, 2)

Hilbert second norm square coefficient(0,0) times left coordinate = 8

Hilbert second norm square term(0,0) := mul(8, 2)

Hilbert second norm square term(0,0) = 16

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, -4)

Hilbert second norm square term(0,1) = -16

Hilbert second norm square coefficient(1,0) times left coordinate := mul(2, -4)

Hilbert second norm square coefficient(1,0) times left coordinate = -8

Hilbert second norm square term(1,0) := mul(-8, 2)

Hilbert second norm square term(1,0) = -16

Hilbert second norm square coefficient(1,1) times left coordinate := mul(5, -4)

Hilbert second norm square coefficient(1,1) times left coordinate = -20

Hilbert second norm square term(1,1) := mul(-20, -4)

Hilbert second norm square term(1,1) = 80

Hilbert second norm square complete bilinear sum := sum(16, -16, -16, 80)

Hilbert second norm square complete bilinear sum = 64

Hilbert sum of separate norm squares := sum(4, 64)

Hilbert sum of separate norm squares = 68

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

Hilbert cross term twice inner product = 0

Hilbert expanded square including cross term := sum(68, 0)

Hilbert expanded square including cross term = 68

same inner product=0;norm square of sum=68;separate squares sum=68;twice cross term=0

bilinear expansion=68;orthogonality premise=是;Pythagorean conclusion status=是

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, 0], [2, -4], 0, 68, 68, 是)

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。