Same-Space Hilbert Certificates, Component Binding, Distance and Orthogonality
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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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Declared dimension=2; it must equal the matrix carrier dimension. Give (dimension,matrix,same_inner_product_norm_metric_carrier,finite_dimensional_complete,qualified?,no_compactness_or_separability_claim).
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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
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,y>=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
Real Hilbert space · qf-mod30-hilbert-pythagorean · candidate
Canonical question JSONL · English JSONL · Validation report