Hilbert space (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Hilbert space" in English language version.

Last modified: 19:37:40, 23 July 2026 (UTC)

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von Neumann 1932 von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, Bibcode:1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.
Hermann Weyl (2009), "Mind and nature", Mind and nature: selected writings on philosophy, mathematics, and physics, Princeton University Press, Bibcode:2009mnsw.book.....W
von Neumann (1955) defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l². The convention still persists in most rigorous treatments of quantum mechanics; see for instance Sobrino 1996, Appendix B. von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link). Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, New Jersey: World Scientific Publishing Co. Inc., Bibcode:1996lnrq.book.....S, doi:10.1142/2865, ISBN 978-981-02-2386-1, MR 1626401.

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Marsden 1974, §2.8 Marsden, Jerrold E. (1974), Elementary classical analysis, W. H. Freeman and Co., MR 0357693.
Deitmar 2005, p. 26. Deitmar, Anton (2005), A First Course in Harmonic Analysis, Universitext (2nd ed.), Springer-Verlag, New York, ISBN 0-387-22837-3, MR 2121678.
Largely from the work of Hermann Grassmann, at the urging of August Ferdinand Möbius (Boyer & Merzbach 1991, pp. 584–586). The first modern axiomatic account of abstract vector spaces ultimately appeared in Giuseppe Peano's 1888 account (Grattan-Guinness 2000, §5.2.2; O'Connor & Robertson 1996). Boyer, Carl Benjamin; Merzbach, Uta C (1991), A History of Mathematics (2nd ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8. Grattan-Guinness, Ivor (2000), The search for mathematical roots, 1870–1940, Princeton Paperbacks, Princeton University Press, ISBN 978-0-691-05858-0, MR 1807717. O'Connor, John J.; Robertson, Edmund F. (1996), "Abstract linear spaces", MacTutor History of Mathematics Archive, University of St Andrews.
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964], "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.), Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications, p. 773, ISBN 978-0-486-61272-0, LCCN 64-60036, MR 0167642, LCCN 65-12253
Pedersen 1995, §4.4 Pedersen, Gert (1995), Analysis Now, Graduate Texts in Mathematics, vol. 118, Berlin, New York: Springer-Verlag, ISBN 978-1-4612-6981-6, MR 0971256.
Bachman, Narici & Beckenstein 2000 Bachman, George; Narici, Lawrence; Beckenstein, Edward (2000), Fourier and wavelet analysis, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98899-3, MR 1729490.
von Neumann 1955 von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link).
Weidmann 1980, Theorem 4.8 Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
Weidmann (1980, Exercise 4.11) Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
Weidmann 1980, §4.5 Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
Buttazzo, Giaquinta & Hildebrandt 1998, Theorem 5.17 Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan (1998), One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, vol. 15, The Clarendon Press Oxford University Press, ISBN 978-0-19-850465-8, MR 1694383.
Shubin 1987, §8.1, Proposition 8.1, p. 67. Shubin, M. A. (1987), Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13621-7, MR 0883081.
Hall 2013, p. 187, Lemma 9.33. Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, New York: Springer, doi:10.1007/978-1-4614-7116-5, ISBN 978-1-4614-7115-8, MR 3112817.
Weidmann 1980, §3.4 Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
For the case of finite index sets, see, for instance, Halmos 1957, §5. For infinite index sets, see Weidmann 1980, Theorem 3.6. Halmos, Paul (1957), Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Pub. Co. Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
von Neumann (1955) defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l². The convention still persists in most rigorous treatments of quantum mechanics; see for instance Sobrino 1996, Appendix B. von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link). Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, New Jersey: World Scientific Publishing Co. Inc., Bibcode:1996lnrq.book.....S, doi:10.1142/2865, ISBN 978-981-02-2386-1, MR 1626401.
von Neumann 1955, Theorem 16 von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link).
von Neumann 1955, Theorem 14 von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link).
Kakutani 1939. Kakutani, Shizuo (1939), "Some characterizations of Euclidean space", Japanese Journal of Mathematics, 16: 93–97, doi:10.4099/jjm1924.16.0_93, MR 0000895.
Lindenstrauss & Tzafriri 1971. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10.1007/BF02771592, ISSN 0021-2172, MR 0276734, S2CID 119575718.
A general account of spectral theory in Hilbert spaces can be found in Riesz & Sz.-Nagy (1990). A more sophisticated account in the language of C*-algebras is in Rudin (1973) or Kadison & Ringrose (1997) Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3. Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259 Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0819-1, MR 1468229.
See, for instance, Riesz & Sz.-Nagy (1990, Chapter VI) or Weidmann 1980, Chapter 7. This result was already known to Schmidt (1908) in the case of operators arising from integral kernels. Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3. Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954. Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, doi:10.1007/BF03029116, S2CID 120666844.
Shubin 1987 Shubin, M. A. (1987), Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13621-7, MR 0883081.
Bernau, S. J. (1966), "The spectral theorem for unbounded normal operators", Pacific Journal of Mathematics, 19 (3): 391–406, doi:10.2140/pjm.1966.19.391, MR 0205065

archive.org (Global: 6^th place; English: 6^th place)

Largely from the work of Hermann Grassmann, at the urging of August Ferdinand Möbius (Boyer & Merzbach 1991, pp. 584–586). The first modern axiomatic account of abstract vector spaces ultimately appeared in Giuseppe Peano's 1888 account (Grattan-Guinness 2000, §5.2.2; O'Connor & Robertson 1996). Boyer, Carl Benjamin; Merzbach, Uta C (1991), A History of Mathematics (2nd ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8. Grattan-Guinness, Ivor (2000), The search for mathematical roots, 1870–1940, Princeton Paperbacks, Princeton University Press, ISBN 978-0-691-05858-0, MR 1807717. O'Connor, John J.; Robertson, Edmund F. (1996), "Abstract linear spaces", MacTutor History of Mathematics Archive, University of St Andrews.
Kline 1972, p. 1092 Kline, Morris (1972), Mathematical thought from ancient to modern times, Volume 3 (3rd ed.), Oxford University Press (published 1990), ISBN 978-0-19-506137-6.
Stein 1970 Stein, E (1970), Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, ISBN 978-0-691-08079-6.
Stein & Weiss 1971, §IV.2 Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
Rudin 1973 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973, p. 295 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973, p. 296 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973, p. 297 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
Rudin 1973, pp. 298–299 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259
A general account of spectral theory in Hilbert spaces can be found in Riesz & Sz.-Nagy (1990). A more sophisticated account in the language of C*-algebras is in Rudin (1973) or Kadison & Ringrose (1997) Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3. Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259 Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0819-1, MR 1468229.
Rudin 1973, Theorem 13.30 Rudin, Walter (1973), Functional Analysis, International Series in Pure and Applied Mathematics, vol. 25 (First ed.), New York, NY: McGraw-Hill Science/Engineering/Math, ISBN 9780070542259

axler.net (Global: low place; English: low place)

linear.axler.net

Axler 2024, p. 182, 6A Inner Products and Norms Axler, Sheldon (2024), Linear Algebra Done Right (PDF), Undergraduate Texts in Mathematics (4th ed.), Springer Publishing, p. 296.
Axler (2024), pp. 183–184. Axler, Sheldon (2024), Linear Algebra Done Right (PDF), Undergraduate Texts in Mathematics (4th ed.), Springer Publishing, p. 296.
Axler (2024), p. 185, Properties 6(d) and 6(e). Axler, Sheldon (2024), Linear Algebra Done Right (PDF), Undergraduate Texts in Mathematics (4th ed.), Springer Publishing, p. 296.
Axler (2024), p. 186. Axler, Sheldon (2024), Linear Algebra Done Right (PDF), Undergraduate Texts in Mathematics (4th ed.), Springer Publishing, p. 296.

books.google.com (Global: 3^rd place; English: 3^rd place)

Lebesgue 1904. Further details on the history of integration theory can be found in Bourbaki (1987) and Saks (2005). Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars. Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9. Saks, Stanisław (2005), Theory of the integral (2nd Dover ed.), Dover, ISBN 978-0-486-44648-6; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.
A treatment of Fourier series from this point of view is available, for instance, in Rudin (1987) or Folland (2009). Rudin, Walter (1987), Real and Complex Analysis, McGraw-Hill, ISBN 978-0-07-100276-9. Folland, Gerald B. (2009), Fourier analysis and its application (Reprint of Wadsworth and Brooks/Cole 1992 ed.), American Mathematical Society Bookstore, ISBN 978-0-8218-4790-9.
Lanczos 1988, pp. 212–213 Lanczos, Cornelius (1988), Applied analysis (Reprint of 1956 Prentice-Hall ed.), Dover Publications, ISBN 978-0-486-65656-4.
Lanczos 1988, Equation 4-3.10 Lanczos, Cornelius (1988), Applied analysis (Reprint of 1956 Prentice-Hall ed.), Dover Publications, ISBN 978-0-486-65656-4.
Aliprantis & Border 2006, p. 250 Aliprantis, Charalambos D.; Border, Kim C. (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.), Springer.
Weisenburger, Steven (2006), A Gravity's Rainbow Companion: Sources and Contexts for Pynchon's Novel (2nd ed.), University of Georgia Press, p. 139, ISBN 9780820337647

doi.org (Global: 2^nd place; English: 2^nd place)

Olver, Peter J.; Shakiban, Chehrzad (2018), Applied Linear Algebra, Undergraduate Texts in Mathematics, Springer International Publishing, p. 137, doi:10.1007/978-3-319-91041-3, ISBN 9783319910413
Kainth (2023). For the completeness of Euclidean space, see Definition 4.37 and Example 4.38, p. 108; for the equivalence of completeness with the property that absolutely convergent series converge, see Theorem 4.44, p. 110. Kainth, Surinder Pal Singh (2023), A Comprehensive Textbook on Metric Spaces, Springer Nature Singapore, doi:10.1007/978-981-99-2738-8, ISBN 9789819927388.
Vince, John (2018), Imaginary Mathematics for Computer Science, Springer International Publishing, doi:10.1007/978-3-319-94637-5, ISBN 9783319946375
Kainth (2023), p. 108, Definition 4.37. Kainth, Surinder Pal Singh (2023), A Comprehensive Textbook on Metric Spaces, Springer Nature Singapore, doi:10.1007/978-981-99-2738-8, ISBN 9789819927388.
Schmidt 1908 Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, doi:10.1007/BF03029116, S2CID 120666844.
von Neumann 1929 von Neumann, John (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische Annalen, 102: 49–131, doi:10.1007/BF01782338, S2CID 121249803.
Hilbert, Nordheim & von Neumann 1927 Hilbert, David; Nordheim, Lothar Wolfgang; von Neumann, John (1927), "Über die Grundlagen der Quantenmechanik", Mathematische Annalen, 98: 1–30, doi:10.1007/BF01451579, S2CID 120986758.
von Neumann 1932 von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, Bibcode:1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.
Kac 1966. Kac, Mark (1966), "Can one hear the shape of a drum?", American Mathematical Monthly, 73 (4, part 2): 1–23, doi:10.2307/2313748, JSTOR 2313748.
Berthier, M. (2020), "Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit", The Journal of Mathematical Neuroscience, 10 (1) 14, doi:10.1186/s13408-020-00092-x, PMC 7481323, PMID 32902776
Clarkson 1936 Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc., 40 (3): 396–414, doi:10.2307/1989630, JSTOR 1989630.
Atiyah & Singer 1968. Atiyah, Michael F.; Singer, Isadore M. (1968), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715.
Hall 2013, p. 187, Lemma 9.33. Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, New York: Springer, doi:10.1007/978-1-4614-7116-5, ISBN 978-1-4614-7115-8, MR 3112817.
von Neumann (1955) defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l². The convention still persists in most rigorous treatments of quantum mechanics; see for instance Sobrino 1996, Appendix B. von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., Princeton University Press, ISBN 978-0-691-02893-4, MR 1435976{{citation}}: CS1 maint: ignored ISBN errors (link). Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, New Jersey: World Scientific Publishing Co. Inc., Bibcode:1996lnrq.book.....S, doi:10.1142/2865, ISBN 978-981-02-2386-1, MR 1626401.
Kakutani 1939. Kakutani, Shizuo (1939), "Some characterizations of Euclidean space", Japanese Journal of Mathematics, 16: 93–97, doi:10.4099/jjm1924.16.0_93, MR 0000895.
Lindenstrauss & Tzafriri 1971. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10.1007/BF02771592, ISSN 0021-2172, MR 0276734, S2CID 119575718.
See, for instance, Riesz & Sz.-Nagy (1990, Chapter VI) or Weidmann 1980, Chapter 7. This result was already known to Schmidt (1908) in the case of operators arising from integral kernels. Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3. Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954. Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, doi:10.1007/BF03029116, S2CID 120666844.
Bernau, S. J. (1966), "The spectral theorem for unbounded normal operators", Pacific Journal of Mathematics, 19 (3): 391–406, doi:10.2140/pjm.1966.19.391, MR 0205065

encyclopediaofmath.org (Global: 7,049^th place; English: 5,456^th place)

However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g., Levitan 2001. Levitan, B.M. (2001) [1994], "Hilbert space", Encyclopedia of Mathematics, EMS Press.
Levitan 2001. Many authors, such as Dunford & Schwartz (1958, §IV.4), refer to this just as the dimension. Unless the Hilbert space is finite dimensional, this is not the same thing as its dimension as a linear space (the cardinality of a Hamel basis). Levitan, B.M. (2001) [1994], "Hilbert space", Encyclopedia of Mathematics, EMS Press. Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience.

ethz.ch (Global: 2,203^rd place; English: 1,928^th place)

people.math.ethz.ch

Kowalski, Emmanuel, "Spectral theory in Hilbert spaces" (PDF)

jstor.org (Global: 23^rd place; English: 15^th place)

von Neumann 1932 von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, Bibcode:1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.
Kac 1966. Kac, Mark (1966), "Can one hear the shape of a drum?", American Mathematical Monthly, 73 (4, part 2): 1–23, doi:10.2307/2313748, JSTOR 2313748.
Clarkson 1936 Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc., 40 (3): 396–414, doi:10.2307/1989630, JSTOR 1989630.
Atiyah & Singer 1968. Atiyah, Michael F.; Singer, Isadore M. (1968), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715.

loc.gov (Global: 63^rd place; English: 47^th place)

lccn.loc.gov

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964], "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.), Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications, p. 773, ISBN 978-0-486-61272-0, LCCN 64-60036, MR 0167642, LCCN 65-12253

loc.gov

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964], "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.), Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications, p. 773, ISBN 978-0-486-61272-0, LCCN 64-60036, MR 0167642, LCCN 65-12253

nih.gov (Global: 5^th place; English: 5^th place)

ncbi.nlm.nih.gov

von Neumann 1932 von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, Bibcode:1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.
Berthier, M. (2020), "Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit", The Journal of Mathematical Neuroscience, 10 (1) 14, doi:10.1186/s13408-020-00092-x, PMC 7481323, PMID 32902776

pubmed.ncbi.nlm.nih.gov

von Neumann 1932 von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, Bibcode:1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674.
Berthier, M. (2020), "Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit", The Journal of Mathematical Neuroscience, 10 (1) 14, doi:10.1186/s13408-020-00092-x, PMC 7481323, PMID 32902776

nii.ac.jp (Global: 195^th place; English: 1,046^th place)

Koashi, Masato, "Appendix: Linear algebra" (PDF)

ox.ac.uk (Global: 541^st place; English: 356^th place)

people.maths.ox.ac.uk

Green, Ben, "A Brief Primer on Conditional Expectation" (PDF)

pitt.edu (Global: 1,616^th place; English: 1,079^th place)

philsci-archive.pitt.edu

Earman 2020. Earman, John (2020), Quantum Physics in Non-Separable Hilbert Spaces.

projecteuclid.org (Global: 4,781^st place; English: 3,117^th place)

Bernau, S. J. (1966), "The spectral theorem for unbounded normal operators", Pacific Journal of Mathematics, 19 (3): 391–406, doi:10.2140/pjm.1966.19.391, MR 0205065

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

Schmidt 1908 Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, doi:10.1007/BF03029116, S2CID 120666844.
von Neumann 1929 von Neumann, John (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische Annalen, 102: 49–131, doi:10.1007/BF01782338, S2CID 121249803.
Hilbert, Nordheim & von Neumann 1927 Hilbert, David; Nordheim, Lothar Wolfgang; von Neumann, John (1927), "Über die Grundlagen der Quantenmechanik", Mathematische Annalen, 98: 1–30, doi:10.1007/BF01451579, S2CID 120986758.
Lindenstrauss & Tzafriri 1971. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10.1007/BF02771592, ISSN 0021-2172, MR 0276734, S2CID 119575718.
See, for instance, Riesz & Sz.-Nagy (1990, Chapter VI) or Weidmann 1980, Chapter 7. This result was already known to Schmidt (1908) in the case of operators arising from integral kernels. Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3. Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954. Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, doi:10.1007/BF03029116, S2CID 120666844.

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

mathshistory.st-andrews.ac.uk

Largely from the work of Hermann Grassmann, at the urging of August Ferdinand Möbius (Boyer & Merzbach 1991, pp. 584–586). The first modern axiomatic account of abstract vector spaces ultimately appeared in Giuseppe Peano's 1888 account (Grattan-Guinness 2000, §5.2.2; O'Connor & Robertson 1996). Boyer, Carl Benjamin; Merzbach, Uta C (1991), A History of Mathematics (2nd ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8. Grattan-Guinness, Ivor (2000), The search for mathematical roots, 1870–1940, Princeton Paperbacks, Princeton University Press, ISBN 978-0-691-05858-0, MR 1807717. O'Connor, John J.; Robertson, Edmund F. (1996), "Abstract linear spaces", MacTutor History of Mathematics Archive, University of St Andrews.

ubc.ca (Global: 1,820^th place; English: 998^th place)

math.ubc.ca

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964], "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.), Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications, p. 773, ISBN 978-0-486-61272-0, LCCN 64-60036, MR 0167642, LCCN 65-12253

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mat.univie.ac.at

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search.worldcat.org

Schaefer & Wolff 1999, pp. 122–202 Schaefer, Helmut H.; Wolff, Manfred P. (1999), Topological Vector Spaces, GTM, vol. 8 (Second ed.), New York, NY: Springer New York Imprint Springer, ISBN 978-1-4612-7155-0, OCLC 840278135
Schaefer & Wolff 1999, pp. 40–41. Schaefer, Helmut H.; Wolff, Manfred P. (1999), Topological Vector Spaces, GTM, vol. 8 (Second ed.), New York, NY: Springer New York Imprint Springer, ISBN 978-1-4612-7155-0, OCLC 840278135
Peres 1993, pp. 79–99 Peres, Asher (1993), Quantum Theory: Concepts and Methods, Kluwer, ISBN 0-7923-2549-4, OCLC 28854083.
Holevo 2001, p. 17 Holevo, Alexander S. (2001), Statistical Structure of Quantum Theory, Lecture Notes in Physics, Springer, ISBN 3-540-42082-7, OCLC 318268606.
Peres 1993, p. 101 Peres, Asher (1993), Quantum Theory: Concepts and Methods, Kluwer, ISBN 0-7923-2549-4, OCLC 28854083.
Peres 1993, pp. 73 Peres, Asher (1993), Quantum Theory: Concepts and Methods, Kluwer, ISBN 0-7923-2549-4, OCLC 28854083.
Nielsen & Chuang 2000, p. 90 Nielsen, Michael A.; Chuang, Isaac L. (2000), Quantum Computation and Quantum Information (1st ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-63503-5, OCLC 634735192.
Peres 1993, pp. 77–78 Peres, Asher (1993), Quantum Theory: Concepts and Methods, Kluwer, ISBN 0-7923-2549-4, OCLC 28854083.
Schaefer & Wolff 1999, pp. 185–190. Schaefer, Helmut H.; Wolff, Manfred P. (1999), Topological Vector Spaces, GTM, vol. 8 (Second ed.), New York, NY: Springer New York Imprint Springer, ISBN 978-1-4612-7155-0, OCLC 840278135
Lindenstrauss & Tzafriri 1971. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10.1007/BF02771592, ISSN 0021-2172, MR 0276734, S2CID 119575718.

zbmath.org (Global: 2,089^th place; English: 1,141^st place)

Young 1988, Chapter 9 Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33071-8, Zbl 0645.46024.
Young 1988, p. 23 Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33071-8, Zbl 0645.46024.
Young 1988, Theorem 15.3 Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33071-8, Zbl 0645.46024.

Hilbert space (English Wikipedia)

adsabs.harvard.edu (Global: 14th place; English: 9th place)

ui.adsabs.harvard.edu

ams.org (Global: 434th place; English: 249th place)

mathscinet.ams.org

archive.org (Global: 6th place; English: 6th place)

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ethz.ch (Global: 2,203rd place; English: 1,928th place)