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      "statement": "# Riesz and Fischer: a home for limits measured by average squared error\n\n## Can increasingly accurate approximations fail to have a home?\n\nSuppose a sequence of curves becomes increasingly close in average squared error. Its late members agree with each other, but does a legitimate limiting curve exist? A space is complete when such Cauchy sequences always have a limit inside the space. Without completeness, an approximation procedure could appear to settle while its supposed answer lies outside the allowed objects. Fischer establishes this property for square-integrable real functions on a finite interval, with functions equal except on a zero-measure set identified.\n\n## Average squared error is one particular ruler\n\nFor two functions, subtract their values, square the difference and integrate over the interval. A small result means the total squared error is small. It need not mean every point has a small error. In a classroom example, a patch of height one and width one divided by n has squared error one divided by n from zero. The error tends to zero even though the largest error remains one. The choice of ruler changes what convergence promises.\n\n## The infinite-coordinate bridge\n\nOrthogonal functions behave like perpendicular coordinate directions. Riesz shows that any list of coefficients whose squared magnitudes have a finite sum can be realized by a square-integrable function against a normalized orthogonal sequence. Fischer independently derives the corresponding realization from completeness. With a complete orthogonal system, the coordinates determine the whole function up to zero-measure changes. If some directions are missing, an invisible orthogonal-complement component can remain, so uniqueness among all functions is not automatic.\n\n## Two papers, different contributions\n\nThe familiar paired name does not identify a joint article. Riesz publishes one original note, while Fischer gives independent notes and acknowledges publication priority. Riesz also explicitly sketches parts of his argument rather than supplying every detail. Hilbert, Schmidt, Fatou and Lebesgue provide important earlier inputs. The admitted contribution is the original completeness and coordinate-realization family, not ownership of every later abstract Hilbert-space theorem or a proof that arbitrary Fourier series converge at each point.\n\n## Try a check with a clear stopping point\n\nAsk your agent to compute the integral of the squared patch error and its largest error separately. Then let it explain why knowing only the first coordinate of a two-dimensional vector cannot determine the vector. These are teaching analogies, not reconstructions of the original proof. A useful result is a short explanation distinguishing the error metric, completeness of the space and completeness of the coordinate system. Keep those three ideas separate before using an infinite expansion.\n\n## Scientific scope\n\nThe original scope is real Lebesgue-square-integrable functions on a finite interval. Mean-square convergence does not imply pointwise or uniform convergence of the entire sequence. For an incomplete orthogonal system, coefficients determine the closed-span representative but leave an orthogonal-complement component undetermined. The authors wrote independent papers; Fischer acknowledges Riesz publication priority. Riesz explicitly gives a proof sketch in places. Earlier Hilbert, Schmidt, Fatou and Lebesgue results retain their own credit.\n\n## Sources for this explanation\n\n- [Ueber orthogonale Funktionensysteme](https://gdz.sub.uni-goettingen.de/download/PPN252457811_1907/PPN252457811_1907___LOG_0016.pdf): Original printed116–122,orthogonal-coefficient realization and explicit proof-sketch boundary.\n- [Sur la convergence en moyenne —originalpage1022](https://thumb.wikimedia.org/wikipedia/commons/thumb/0/04/Comptes_rendus_hebdomadaires_des_s%C3%A9ances_de_l%E2%80%99Acad%C3%A9mie_des_sciences%2C_tome_144%2C_1907.djvu/page1022-1920px-Comptes_rendus_hebdomadaires_des_s%C3%A9ances_de_l%E2%80%99Acad%C3%A9mie_des_sciences%2C_tome_144%2C_1907.djvu.jpg): First original note pp1022–1024,mean-square completeness and Riesz priority; companion pp1148–1151applies it to orthogonal coefficients.\n",
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    "what_changed": "A sequence of curves can settle down in average squared error without settling at every point. Fischer proves that, on a finite interval, a sequence that is Cauchy in this mean-square sense has a square-integrable mean-square limit. Riesz shows that every square-summable list of coefficients can be realized against an arbitrary normalized orthogonal sequence; Fischer independently obtains this using completeness. Together these results connect functions with infinite lists of coordinates. If the orthogonal system is complete, the correspondence describes the whole function space, with functions differing only on a set of measure zero treated as the same.",
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        "title": "Ueber orthogonale Funktionensysteme",
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        "title": "Can increasingly accurate approximations fail to have a home?",
        "text": "Suppose a sequence of curves becomes increasingly close in average squared error. Its late members agree with each other, but does a legitimate limiting curve exist? A space is complete when such Cauchy sequences always have a limit inside the space. Without completeness, an approximation procedure could appear to settle while its supposed answer lies outside the allowed objects. Fischer establishes this property for square-integrable real functions on a finite interval, with functions equal except on a zero-measure set identified."
      },
      {
        "title": "Average squared error is one particular ruler",
        "text": "For two functions, subtract their values, square the difference and integrate over the interval. A small result means the total squared error is small. It need not mean every point has a small error. In a classroom example, a patch of height one and width one divided by n has squared error one divided by n from zero. The error tends to zero even though the largest error remains one. The choice of ruler changes what convergence promises."
      },
      {
        "title": "The infinite-coordinate bridge",
        "text": "Orthogonal functions behave like perpendicular coordinate directions. Riesz shows that any list of coefficients whose squared magnitudes have a finite sum can be realized by a square-integrable function against a normalized orthogonal sequence. Fischer independently derives the corresponding realization from completeness. With a complete orthogonal system, the coordinates determine the whole function up to zero-measure changes. If some directions are missing, an invisible orthogonal-complement component can remain, so uniqueness among all functions is not automatic."
      },
      {
        "title": "Two papers, different contributions",
        "text": "The familiar paired name does not identify a joint article. Riesz publishes one original note, while Fischer gives independent notes and acknowledges publication priority. Riesz also explicitly sketches parts of his argument rather than supplying every detail. Hilbert, Schmidt, Fatou and Lebesgue provide important earlier inputs. The admitted contribution is the original completeness and coordinate-realization family, not ownership of every later abstract Hilbert-space theorem or a proof that arbitrary Fourier series converge at each point."
      },
      {
        "title": "Try a check with a clear stopping point",
        "text": "Ask your agent to compute the integral of the squared patch error and its largest error separately. Then let it explain why knowing only the first coordinate of a two-dimensional vector cannot determine the vector. These are teaching analogies, not reconstructions of the original proof. A useful result is a short explanation distinguishing the error metric, completeness of the space and completeness of the coordinate system. Keep those three ideas separate before using an infinite expansion."
      }
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        "title": "Ueber orthogonale Funktionensysteme",
        "url": "https://gdz.sub.uni-goettingen.de/download/PPN252457811_1907/PPN252457811_1907___LOG_0016.pdf",
        "locator": "Original printed116–122,orthogonal-coefficient realization and explicit proof-sketch boundary."
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        "title": "Sur la convergence en moyenne —originalpage1022",
        "url": "https://thumb.wikimedia.org/wikipedia/commons/thumb/0/04/Comptes_rendus_hebdomadaires_des_s%C3%A9ances_de_l%E2%80%99Acad%C3%A9mie_des_sciences%2C_tome_144%2C_1907.djvu/page1022-1920px-Comptes_rendus_hebdomadaires_des_s%C3%A9ances_de_l%E2%80%99Acad%C3%A9mie_des_sciences%2C_tome_144%2C_1907.djvu.jpg",
        "locator": "First original note pp1022–1024,mean-square completeness and Riesz priority; companion pp1148–1151applies it to orthogonal coefficients."
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