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R0029-15258 · q-20261014-advance-20261030-r02-real-hilbert-space-0008
Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product <x,y>=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=[[9, 1], [1, 2]] and inner product <x,y>=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).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.
Before
Hilbert Gram diagonal product := mul(9, 2)
After
Hilbert Gram diagonal product = 18
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Hilbert Gram determinant := sub(18, 1)
After
Hilbert Gram determinant = 17
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Hilbert completing-square slope := div(1, 9)
After
Hilbert completing-square slope = 1/9
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Hilbert completing-square positive residual := div(17, 9)
After
Hilbert completing-square positive residual = 17/9
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
complete real carrier R^2;Gram matrix=[[9, 1], [1, 2]];a=9>0;det=17>0;dimension declaration=2
After
x^T M x=9*(x0+(1/9)*x1)^2+(17/9)*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.
Rule: hilbert_same_complete_matrix_positive_inner_product · Uses: G001, S005
Before
same R^2 carrier and complete Gram matrix=[[9, 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.
Rule: hilbert_named_certificate_same_norm_metric_completeness · Uses: G001, S006
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.(2, [[9, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: d6d0c13be09b1e4c35058d4cf02057f1fb58cd4faaac1e6db6a4a08fbe907d75candidate record SHA-256: 0929ec3c652234fab777ba79707784149ea6397900fe311d21b6d6b4497df894semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117Lean: NOT_RUN · human review: NOT_PERFORMED