{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "schur-weak-to-norm-l1-matrix-transformations-1921",
    "title": "Schur: when testing against every bounded sequence forces norm convergence",
    "summary": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm.",
    "boundary": "Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.",
    "area": "Functional analysis and infinite matrix summability",
    "topic": "Mathematics",
    "year": 1921,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "schur-weak-to-norm-l1-matrix-transformations-1921",
    "target": {
      "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
      "version": 1,
      "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
    },
    "humans": [
      "J. Schur"
    ],
    "authors_complete": true,
    "human_role_note": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "schur-linear-transformations-original1921-p79",
        "title": "Schur1921original JRAM151 printedp79",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p80",
        "title": "Schur1921original JRAM151 printedp80",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000086.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p81",
        "title": "Schur1921original JRAM151 printedp81",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000087.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p82",
        "title": "Schur1921original JRAM151 printedp82",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000088.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p83",
        "title": "Schur1921original JRAM151 printedp83",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000089.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p84",
        "title": "Schur1921original JRAM151 printedp84",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000090.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p85",
        "title": "Schur1921original JRAM151 printedp85",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000091.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p86",
        "title": "Schur1921original JRAM151 printedp86",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000092.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p87",
        "title": "Schur1921original JRAM151 printedp87",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000093.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p88",
        "title": "Schur1921original JRAM151 printedp88",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000094.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p89",
        "title": "Schur1921original JRAM151 printedp89",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000095.jpg",
        "date": "1921"
      },
      {
        "id": "schur-linear-transformations-original1921-p90",
        "title": "Schur1921original JRAM151 printedp90",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000096.jpg",
        "date": "1921"
      }
    ],
    "geography": {
      "country_codes": [
        "DE"
      ],
      "location_basis": "Original byline gives Berlin. Present-day place labels do not assert nationality,modern-state ownership or historical territorial jurisdiction.",
      "source_url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg",
      "target": {
        "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
        "version": 1,
        "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
      },
      "basis": "RESEARCH_LOCATION_OR_PAPER_AFFILIATION",
      "reviewed_on": "2026-10-23",
      "nationality_inferred": false,
      "sources": [
        {
          "title": "Schur1921original JRAM151 printedp79",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg"
        }
      ],
      "coverage_status": "COMPLETE",
      "country_labels": {
        "DE": "Germany"
      }
    },
    "points": "250000",
    "published_grade": {
      "id": "schur-weak-to-norm-l1-matrix-transformations-1921",
      "entry_id": "schur-weak-to-norm-l1-matrix-transformations-1921",
      "band": "A",
      "modifier": "+",
      "target": {
        "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
        "version": 1,
        "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
      },
      "what_changed": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm.",
      "scope": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm. Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
        "version": 1,
        "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "J. Schur",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Schur1921original JRAM151 printedp79",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp79",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp80",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000086.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-2",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp81",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000087.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-3",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp82",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000088.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-4",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp83",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000089.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-5",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp84",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000090.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-6",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp85",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000091.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-7",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp86",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000092.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-8",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp87",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000093.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-9",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp88",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000094.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-10",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp89",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000095.jpg"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-11",
          "kind": "source_reference",
          "source": {
            "citation": "Schur1921original JRAM151 printedp90",
            "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000096.jpg"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.",
        "domain": "Functional analysis and infinite matrix summability"
      },
      "sources": [
        {
          "citation": "Schur1921original JRAM151 printedp79",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp80",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000086.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp81",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000087.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp82",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000088.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp83",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000089.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp84",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000090.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp85",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000091.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp86",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000092.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp87",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000093.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp88",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000094.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp89",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000095.jpg"
        },
        {
          "citation": "Schur1921original JRAM151 printedp90",
          "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000096.jpg"
        }
      ],
      "statement": "# Schur: when testing against every bounded sequence forces norm convergence\n\n## Every fixed coordinate can settle while the total stays put\n\nConsider an infinite list that has a single one in position n and zeros elsewhere. As n grows, every fixed coordinate eventually becomes zero. Yet the sum of absolute values remains one. This simple example warns that coordinate-by-coordinate convergence can miss mass moving farther out in a list. Schur studies exactly the kind of tail control needed when infinite matrices are used to transform every bounded input sequence into a convergent output.\n\n## Test the whole row, not just its first few entries\n\nEach matrix row combines the input values with that row's weights. Coordinatewise limits of the weights are necessary, but insufficient. Schur requires the absolute tails to become uniformly small. Equivalently, the absolute row sums must converge to the absolute sum of the limiting row. If too much weight can keep escaping to later coordinates, the proof chooses one bounded input whose signs make the row outputs oscillate. That input exposes what separate-coordinate checks cannot see.\n\n## The modern weak-to-norm interpretation\n\nThe space l1 consists of numerical sequences whose absolute values have a finite sum. Weak convergence tests a sequence of such objects against every bounded scalar sequence. That is much stronger than testing each coordinate individually. Schur's result implies that a weakly convergent sequence in l1 actually converges in the absolute-sum norm. It is a special structural fact about this sequence space, not a rule that all weak convergence in all spaces is norm convergence.\n\n## The moving one fails the stronger test\n\nFor the classroom sequence with one at position n, choose a fixed bounded testing sequence with entries alternating between plus one and minus one. Pairing the two gives alternating outputs. Thus the moving-one sequence is coordinatewise null but not weakly null. This explains why it is not a counterexample to Schur's result. A single bounded test can invalidate weak convergence, whereas claiming weak convergence requires handling all allowed tests.\n\n## Preserve the exact contribution when sharing it\n\nAsk your agent to calculate the coordinate limits, absolute-sum norms and alternating-test outputs for this example. It should distinguish convergence preservation from regular summation, which also preserves the original limit. Toeplitz and Steinhaus retain the earlier results explicitly credited in the paper. Other theorems named Schur, including entrywise matrix products and unitary triangularization, are different contributions. A good next step is a clear example with these boundaries, not a statement that weak and norm topologies are identical.\n\n## Scientific scope\n\nCoordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.\n\n## Sources for this explanation\n\n- [Schur1921original JRAM151 printedp79](https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg): Original pp79–90,definitions,TheoremsI–IV,uniform tails and oscillating bounded-test construction.\n",
      "summary": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm.",
      "title": "Schur: when testing against every bounded sequence forces norm convergence"
    },
    "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
      "version": 1,
      "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:bd6e90d475c8e1975c389f3167465debc488fff7c3451d35c9a33577e8f39b63"
  },
  "target": {
    "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
    "version": 1,
    "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "schur-weak-to-norm-l1-matrix-transformations-1921",
    "entry_id": "schur-weak-to-norm-l1-matrix-transformations-1921",
    "revision": 1,
    "title": "Schur: when testing against every bounded sequence forces norm convergence",
    "context": "Functional analysis and infinite matrix summability",
    "track": "Exact theorem and reusable mathematical method",
    "band": "A",
    "modifier": "+",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm.",
    "scope": "An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm. Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.",
    "references": [
      {
        "title": "Schur1921original JRAM151 printedp79",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp80",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000086.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp81",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000087.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp82",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000088.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp83",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000089.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp84",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000090.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp85",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000091.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp86",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000092.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp87",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000093.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp88",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000094.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp89",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000095.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Schur1921original JRAM151 printedp90",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000096.jpg",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
      "version": 1,
      "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "250000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "schur-weak-to-norm-l1-matrix-transformations-1921",
    "target": {
      "record_id": "urn:uuid:a8918743-e46c-47ec-a4fe-3af3687e3ef3",
      "version": 1,
      "content_hash": "sha256:13f0fdd8f63c10cb3ddbccb7dc0a694929193127117407517e255e097f3d53fa"
    },
    "sections": [
      {
        "title": "Every fixed coordinate can settle while the total stays put",
        "text": "Consider an infinite list that has a single one in position n and zeros elsewhere. As n grows, every fixed coordinate eventually becomes zero. Yet the sum of absolute values remains one. This simple example warns that coordinate-by-coordinate convergence can miss mass moving farther out in a list. Schur studies exactly the kind of tail control needed when infinite matrices are used to transform every bounded input sequence into a convergent output."
      },
      {
        "title": "Test the whole row, not just its first few entries",
        "text": "Each matrix row combines the input values with that row's weights. Coordinatewise limits of the weights are necessary, but insufficient. Schur requires the absolute tails to become uniformly small. Equivalently, the absolute row sums must converge to the absolute sum of the limiting row. If too much weight can keep escaping to later coordinates, the proof chooses one bounded input whose signs make the row outputs oscillate. That input exposes what separate-coordinate checks cannot see."
      },
      {
        "title": "The modern weak-to-norm interpretation",
        "text": "The space l1 consists of numerical sequences whose absolute values have a finite sum. Weak convergence tests a sequence of such objects against every bounded scalar sequence. That is much stronger than testing each coordinate individually. Schur's result implies that a weakly convergent sequence in l1 actually converges in the absolute-sum norm. It is a special structural fact about this sequence space, not a rule that all weak convergence in all spaces is norm convergence."
      },
      {
        "title": "The moving one fails the stronger test",
        "text": "For the classroom sequence with one at position n, choose a fixed bounded testing sequence with entries alternating between plus one and minus one. Pairing the two gives alternating outputs. Thus the moving-one sequence is coordinatewise null but not weakly null. This explains why it is not a counterexample to Schur's result. A single bounded test can invalidate weak convergence, whereas claiming weak convergence requires handling all allowed tests."
      },
      {
        "title": "Preserve the exact contribution when sharing it",
        "text": "Ask your agent to calculate the coordinate limits, absolute-sum norms and alternating-test outputs for this example. It should distinguish convergence preservation from regular summation, which also preserves the original limit. Toeplitz and Steinhaus retain the earlier results explicitly credited in the paper. Other theorems named Schur, including entrywise matrix products and unitary triangularization, are different contributions. A good next step is a clear example with these boundaries, not a statement that weak and norm topologies are identical."
      }
    ],
    "references": [
      {
        "title": "Schur1921original JRAM151 printedp79",
        "url": "https://gdz.sub.uni-goettingen.de/content/PPN243919689_0151/1000/0/00000085.jpg",
        "locator": "Original pp79–90,definitions,TheoremsI–IV,uniform tails and oscillating bounded-test construction."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}