# Same-Space Hilbert Certificates, Component Binding, Distance and Orthogonality This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately. batch-20261014-advance-20261030-r02-real-hilbert-space-foundations · 2026-10-14 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. ## 1. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] 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). View answer and explanation(2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[2, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0001/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 2. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[3, 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). View answer and explanation(2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[3, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0002/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 3. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] 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). View answer and explanation(2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[4, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0003/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 4. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] 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). View answer and explanation(2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[5, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0004/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 5. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[6, 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). View answer and explanation(2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[6, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0005/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 6. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] 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). View answer and explanation(2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[7, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0006/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 7. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] 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). View answer and explanation(2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[8, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0007/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 8. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[9, 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). View answer and explanation(2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[9, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0008/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 9. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] 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). View answer and explanation(2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[10, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0009/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 10. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] 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). View answer and explanation(2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[11, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0010/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 11. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[12, 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). View answer and explanation(2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[12, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0011/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 12. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] 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). View answer and explanation(2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[13, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0012/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 13. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] 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). View answer and explanation(2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[14, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0013/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 14. Bind Finite-Dimensional Completeness to the Same Inner-Product Space 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). View answer and explanation(2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[15, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0014/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 15. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] 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). View answer and explanation(2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[16, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0015/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 16. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] 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). View answer and explanation(2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[17, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0016/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 17. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[18, 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). View answer and explanation(2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[18, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0017/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 18. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] 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). View answer and explanation(2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[19, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0018/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 19. Bind Finite-Dimensional Completeness to the Same Inner-Product Space 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. 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). View answer and explanation(2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[20, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0019/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 20. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[21, 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). View answer and explanation(2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[21, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0020/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 21. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] 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). View answer and explanation(2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[22, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0021/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 22. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] 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). View answer and explanation(2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[23, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0022/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 23. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[24, 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). View answer and explanation(2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[24, 1], [1, 2]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0023/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 24. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] 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). View answer and explanation(2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[25, 2], [2, 5]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0024/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 25. Bind Finite-Dimensional Completeness to the Same Inner-Product Space Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] 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). View answer and explanation(2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim) The finite-dimensional real inner-product completeness theorem is invoked conditionally for this explicit carrier, positive-definite form, induced norm and induced metric. The calculation does not prove the general theorem, and does not infer compactness, separability or an infinite-dimensional result. The exact result, retaining the requested order, is (2, [[26, 0], [0, 1]], same_inner_product_norm_metric_carrier, finite_dimensional_complete, Yes, no_compactness_or_separability_claim). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0025/) Real Hilbert space · qf-mod30-hilbert-named-certificate · candidate ## 26. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[2, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0026/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 27. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[3, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0027/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 28. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[4, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0028/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 29. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[5, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0029/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 30. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[6, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0030/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 31. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[7, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0031/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 32. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[8, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0032/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 33. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[9, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0033/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 34. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[10, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0034/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 35. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[11, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0035/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 36. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, No) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[12, 1], [1, 2]], other, same_norm, finite_dimensional_theorem, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0036/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 37. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[13, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0037/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 38. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[14, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0038/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 39. A Completeness Checklist Must Use One Carrier, Norm and Metric 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. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[15, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0039/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 40. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[16, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0040/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 41. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[17, 0], [0, 1]], other, same_norm, finite_dimensional_theorem, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0041/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 42. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[18, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0042/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 43. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[19, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0043/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 44. A Completeness Checklist Must Use One Carrier, Norm and Metric 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. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[20, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0044/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 45. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[21, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0045/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 46. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=other, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[22, 2], [2, 5]], other, same_norm, finite_dimensional_theorem, No). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0046/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 47. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[23, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0047/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 48. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[24, 1], [1, 2]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0048/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 49. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0049/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 50. A Completeness Checklist Must Use One Carrier, Norm and Metric Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?). View answer and explanation(2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes) The last flag certifies exactly the declared same_inner_product, same_norm and finite_dimensional_theorem chain. A No means this proposed checklist did not supply that certificate; it is not a proof that every other norm or the underlying vector space cannot be Hilbert. The exact result, retaining the requested order, is (2, [[26, 0], [0, 1]], same_inner_product, same_norm, finite_dimensional_theorem, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0050/) Real Hilbert space · qf-mod30-hilbert-component-checklist · candidate ## 51. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[2, 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=[-12, -2], right vector=[12, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-12, -2], [12, -3], [-24, 1], 1153, sqrt(1153)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0051/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 52. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-11, -1], right vector=[11, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-11, -1], [11, -2], [-22, 1], 1410, sqrt(1410)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0052/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 53. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-10, 0], right vector=[10, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-10, 0], [10, -1], [-20, 1], 1525, sqrt(1525)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0053/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 54. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[5, 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=[-9, 1], right vector=[9, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-9, 1], [9, 0], [-18, 1], 1621, sqrt(1621)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0054/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 55. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-8, 2], right vector=[8, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-8, 2], [8, 1], [-16, 1], 1506, sqrt(1506)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0055/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 56. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-7, -2], right vector=[7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0056/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 57. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[8, 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, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-6, -1], [6, 3], [-12, -4], 1168, sqrt(1168)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0057/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 58. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-5, 0], right vector=[5, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-5, 0], [5, -3], [-10, 3], 858, sqrt(858)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-5, 0], [5, -3], [-10, 3], 858, sqrt(858)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0058/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 59. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-4, 1], right vector=[4, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-4, 1], [4, -2], [-8, 3], 589, sqrt(589)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-4, 1], [4, -2], [-8, 3], 589, sqrt(589)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0059/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 60. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[11, 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=[-3, 2], right vector=[3, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-3, 2], [3, -1], [-6, 3], 405, sqrt(405)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-3, 2], [3, -1], [-6, 3], 405, sqrt(405)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0060/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 61. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-2, -2], right vector=[2, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-2, -2], [2, 0], [-4, -2], 216, sqrt(216)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-2, -2], [2, 0], [-4, -2], 216, sqrt(216)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0061/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 62. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-1, -1], right vector=[1, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([-1, -1], [1, 1], [-2, -2], 88, sqrt(88)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([-1, -1], [1, 1], [-2, -2], 88, sqrt(88)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0062/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 63. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[14, 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=[0, 0], right vector=[0, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([0, 0], [0, 2], [0, -2], 4, sqrt(4)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([0, 0], [0, 2], [0, -2], 4, sqrt(4)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0063/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 64. Compute Distance in the Norm Induced by This Positive-Definite Form 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. Left vector=[1, 1], right vector=[-1, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([1, 1], [-1, 3], [2, -2], 60, sqrt(60)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([1, 1], [-1, 3], [2, -2], 60, sqrt(60)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0064/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 65. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[2, 2], right vector=[-2, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([2, 2], [-2, -3], [4, 5], 461, sqrt(461)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([2, 2], [-2, -3], [4, 5], 461, sqrt(461)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0065/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 66. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[17, 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=[3, -2], right vector=[-3, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([3, -2], [-3, -2], [6, 0], 612, sqrt(612)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([3, -2], [-3, -2], [6, 0], 612, sqrt(612)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0066/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 67. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[4, -1], right vector=[-4, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([4, -1], [-4, -1], [8, 0], 1152, sqrt(1152)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0067/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 68. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[5, 0], right vector=[-5, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([5, 0], [-5, 0], [10, 0], 1900, sqrt(1900)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0068/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 69. Compute Distance in the Norm Induced by This Positive-Definite Form 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). View answer and explanation([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([6, 1], [-6, 1], [12, 0], 2880, sqrt(2880)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0069/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 70. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[7, 2], right vector=[-7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([7, 2], [-7, 2], [14, 0], 4116, sqrt(4116)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0070/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 71. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[8, -2], right vector=[-8, 3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([8, -2], [-8, 3], [16, -5], 5437, sqrt(5437)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0071/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 72. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[23, 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=[9, -1], right vector=[-9, -3]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([9, -1], [-9, -3], [18, 2], 7456, sqrt(7456)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0072/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 73. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[10, 0], right vector=[-10, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([10, 0], [-10, -2], [20, 2], 9688, sqrt(9688)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0073/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 74. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[11, 1], right vector=[-11, -1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([11, 1], [-11, -1], [22, 2], 12296, sqrt(12296)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0074/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 75. Compute Distance in the Norm Induced by This Positive-Definite Form Use the same real plane with symmetric positive-definite matrix G=[[26, 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=[12, 2], right vector=[-12, 0]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root). View answer and explanation([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980)) Subtract the vectors and evaluate the complete quadratic form of that difference. Retain the exact nonnegative squared distance and the source sqrt(square) expression without replacing it by a decimal or a Euclidean norm from a different inner product. The exact result, retaining the requested order, is ([12, 2], [-12, 0], [24, 2], 14980, sqrt(14980)). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0075/) Real Hilbert space · qf-mod30-hilbert-distance · candidate ## 76. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[2, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[0, -2]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [0, -2], 4, 18, 10, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0076/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 77. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[3, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[1, -3]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [1, -3], 5, 28, 18, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0077/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 78. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[4, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[2, -4]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 0], [2, -4], 0, 68, 68, Yes) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [2, -4], 0, 68, 68, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0078/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 79. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[5, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[0, -5]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [0, -5], -5, 21, 31, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0079/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 80. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[6, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[1, -6]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [1, -6], -22, 40, 84, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0080/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 81. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[2, -7]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [2, -7], 62, 360, 236, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0081/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 82. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[8, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[0, -8]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [0, -8], 8, 89, 73, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0082/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 83. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[9, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[1, -9]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 0], [1, -9], 0, 162, 162, Yes) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [1, -9], 0, 162, 162, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0083/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 84. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[10, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[2, -10]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [2, -10], -46, 387, 479, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0084/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 85. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[11, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[0, -11]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [0, -11], -22, 92, 136, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0085/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 86. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[1, -12]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0086/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 87. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[13, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[2, -13]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [2, -13], 61, 929, 807, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0087/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 88. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[14, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[0, -14]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 0], [0, -14], 0, 210, 210, Yes) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [0, -14], 0, 210, 210, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0088/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 89. Check Orthogonality before Using the Inner-Product Pythagorean Identity 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. Vectors x=[1, 1], y=[1, -15]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [1, -15], -29, 396, 454, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0089/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 90. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[2, -16]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0090/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 91. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[17, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[0, -17]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [0, -17], 34, 378, 310, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0091/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 92. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[18, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[1, -18]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [1, -18], 35, 718, 648, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0092/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 93. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[19, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[2, -19]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 0], [2, -19], 0, 1748, 1748, Yes) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [2, -19], 0, 1748, 1748, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0093/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 94. Check Orthogonality before Using the Inner-Product Pythagorean Identity 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. Vectors x=[1, 1], y=[0, -20]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [0, -20], -20, 381, 421, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0094/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 95. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[1, -21]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [1, -21], -82, 730, 894, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0095/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 96. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[22, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[2, -22]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -2], [2, -22], 212, 2790, 2366, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0096/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 97. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[23, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -1], y=[0, -23]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, -1], [0, -23], 23, 599, 553, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0097/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 98. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[24, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 0], y=[1, -24]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 0], [1, -24], 0, 1152, 1152, Yes) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 0], [1, -24], 0, 1152, 1152, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0098/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 99. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 1], y=[2, -25]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 1], [2, -25], -121, 2817, 3059, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0099/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate ## 100. Check Orthogonality before Using the Inner-Product Pythagorean Identity Use the same real plane with symmetric positive-definite matrix G=[[26, 0], [0, 1]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[0, -26]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). View answer and explanation([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false) Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [0, -26], -52, 602, 706, orthogonality_premise_false). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0100/) Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/validation-report.json)