# Question 15244: Compute Distance in the Norm Induced by This Positive-Definite Form [Return to catalogue page 77](https://ocme.evemisslab.com/en/questions/page/77/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/#q-20261014-advance-20261030-r02-real-hilbert-space-0069) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0029-15244 · q-20261014-advance-20261030-r02-real-hilbert-space-0069 ## Original question and givens Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[6, 1], right vector=[-6, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). - G001 — Use the same real plane with symmetric positive-definite matrix G=[[20, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[6, 1], right vector=[-6, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-real-hilbert-space](https://ocme.evemisslab.com/en/mko/mko-real-hilbert-space/) ## Step-by-step derivation S001 · calculationBefore Hilbert Gram diagonal product := mul(20, 1) After Hilbert Gram diagonal product = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore Hilbert Gram off-diagonal product := mul(0, 0) After Hilbert Gram off-diagonal product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Hilbert Gram determinant := sub(20, 0) After Hilbert Gram determinant = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Hilbert completing-square slope := div(0, 20) After Hilbert completing-square slope = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Hilbert completing-square positive residual := div(20, 20) After Hilbert completing-square positive residual = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore complete real carrier R^2;Gram matrix=[[20, 0], [0, 1]];a=20>0;det=20>0;dimension declaration=2 After x^T M x=20*(x0+(0)*x1)^2+(1)*x1^2;zero only at x=0 Conditions: none Symmetry and positive principal minors give positivity for every real vector by completing the square,not just testing the displayed vectors.The triangular transformation is invertible,so this inner-product norm is equivalent to the coordinate norm on R².A Cauchy sequence has convergent coordinates and its limit remains in this same R². Step identifiers and dependenciesRule: hilbert_same_complete_matrix_positive_inner_product · Uses: G001, S005 S007 · calculationBefore Hilbert distance coordinate0 difference := sub(6, -6) After Hilbert distance coordinate0 difference = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Hilbert distance coordinate1 difference := sub(1, 1) After Hilbert distance coordinate1 difference = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Hilbert distance coefficient(0,0) times left coordinate := mul(20, 12) After Hilbert distance coefficient(0,0) times left coordinate = 240 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Hilbert distance term(0,0) := mul(240, 12) After Hilbert distance term(0,0) = 2880 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Hilbert distance coefficient(0,1) times left coordinate := mul(0, 12) After Hilbert distance coefficient(0,1) times left coordinate = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Hilbert distance term(0,1) := mul(0, 0) After Hilbert distance term(0,1) = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Hilbert distance coefficient(1,0) times left coordinate := mul(0, 0) After Hilbert distance coefficient(1,0) times left coordinate = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Hilbert distance term(1,0) := mul(0, 12) After Hilbert distance term(1,0) = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Hilbert distance coefficient(1,1) times left coordinate := mul(1, 0) After Hilbert distance coefficient(1,1) times left coordinate = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Hilbert distance term(1,1) := mul(0, 0) After Hilbert distance term(1,1) = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Hilbert distance complete bilinear sum := sum(2880, 0, 0, 0) After Hilbert distance complete bilinear sum = 2880 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore left=[6, 1];right=[-6, 1];same induced metric on R^2;squared norm=2880 After distance=sqrt(2880) Conditions: none Subtract coordinates,expand the same Gram quadratic form completely,and take its nonnegative square root.Keep the original sqrt display rather than switching to Euclidean distance or a different simplification. Step identifiers and dependenciesRule: hilbert_distance_nonnegative_root_same_metric · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute Hilbert distance complete bilinear sum with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Hilbert distance term(1,1) with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Hilbert distance term(1,0) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Hilbert distance term(0,1) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Hilbert distance term(0,0) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Hilbert distance coordinate1 difference with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Hilbert distance coordinate0 difference with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals. ## Result ([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880)) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 7ef0fff5196164a386e87dbca3d640c1d86f1c526d765faf081887214d87a6bd - candidate record SHA-256: cb7b9092200e52cc048a3370aa253ed3daad3e4a039985295b5dbea0d0539aa3 - semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725 - question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0069.json)