# Question 15264: Bind Finite-Dimensional Completeness to the Same Inner-Product Space [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-0014) 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. R0029-15264 · q-20261014-advance-20261030-r02-real-hilbert-space-0014 ## Original question and givens Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim). - G001 — Use the same real plane with symmetric positive-definite matrix G=[[15, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim). ## 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(15, 2) After Hilbert Gram diagonal product = 30 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(1, 1) After Hilbert Gram off-diagonal product = 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, S001 S003 · calculationBefore Hilbert Gram determinant := sub(30, 1) After Hilbert Gram determinant = 29 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(1, 15) After Hilbert completing-square slope = 1/15 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(29, 15) After Hilbert completing-square positive residual = 29/15 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=[[15, 1], [1, 2]];a=15>0;det=29>0;dimension declaration=2 After x^T M x=15*(x0+(1/15)*x1)^2+(29/15)*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 · impliesBefore same R^2 carrier and complete Gram matrix=[[15, 1], [1, 2]];induced norm=sqrt(x^T M x);distance=norm(x-y) After same_inner_product_norm_metric_carrier;finite_dimensional_complete;no_compactness_or_separability_claim Conditions: none Use this very inner product for the norm and this very norm for the metric.The finite-dimensional completeness statement applies to that same carrier and norm,which supplies Hilbert qualification.This answer adds no compactness or separability claim and neither property replaces completeness. Step identifiers and dependenciesRule: hilbert_named_certificate_same_norm_metric_completeness · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) 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: 94a7e9e63110c71c817844476f3d5f256409a27dd09946b58d2b9847ec33956d - candidate record SHA-256: 962a435de48aa9ed9be889e3a8b5de738c9357b741855c44fe561cf8947749c3 - 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-0014.json)