Hilbert space (English Wikipedia)

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

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Marsden 1974, §2.8 Marsden, Jerrold E. (1974), Elementary classical analysis, W. H. Freeman and Co., MR 0357693.
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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976.
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.
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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976. 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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976.
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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976.
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.

archive.org

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.
A general reference for this section is Rudin (1973), chapter 12. 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.

books.google.com

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.

doi.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.
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.
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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976. 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.

encyclopediaofmath.org

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.

harvard.edu

ui.adsabs.harvard.edu

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.
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 (published 1996), ISBN 978-0-691-02893-4, MR 1435976. 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.

jstor.org

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.

loc.gov

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

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

semanticscholar.org

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

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

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.

worldcat.org

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.
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
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

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.