Ruang Hilbert

Dalam matematik, ruang Hilbert (dinamakan sempena David Hilbert) membenarkan pengitlak kaedah algebra dan kalkulus linear daripada ruang Euclidean dua dimensi dan tiga dimensi kepada ruang yang mungkin mempunyai dimensi tak terhingga. Ruang Hilbert ialah ruang vektor dilengkapi dengan operasi produk dalaman, yang membolehkan mentakrifkan fungsi jarak dan keserenjang (dikenali sebagai ortogonal dalam konteks ini). Tambahan pula, ruang Hilbert adalah lengkap untuk jarak ini, yang bermaksud terdapat had yang mencukupi dalam ruang untuk membolehkan teknik kalkulus digunakan.

Rujukan

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.
Bers, Lipman; John, Fritz; Schechter, Martin (1981), Partial differential equations, American Mathematical Society, ISBN 978-0-8218-0049-2.
Bourbak, Nicolasi (1986), Spectral theories, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-0-201-00767-1.
Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9.
Boyer, Carl Benjamin; Merzbach, Uta C (1991), A History of Mathematics (ed. 2nd), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8.
Brenner, S.; Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods (ed. 2nd), Springer, ISBN 978-0-387-95451-6.
Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan (1998), One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, 15, The Clarendon Press Oxford University Press, ISBN 978-0-19-850465-8, MR 1694383.
Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc., 40 (3): 396–414, doi:10.2307/1989630, JSTOR 1989630.
Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Vol. I, Interscience.
Dieudonné, Jean (1960), Foundations of Modern Analysis, Academic Press.
Dirac, P.A.M. (1930), The Principles of Quantum Mechanics, Oxford: Clarendon Press.
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience.
Duren, P. (1970), Theory of H^p-Spaces, New York: Academic Press.
Folland, Gerald B. (2009), Fourier analysis and its application (ed. Reprint of Wadsworth and Brooks/Cole 1992), American Mathematical Society Bookstore, ISBN 978-0-8218-4790-9.
Folland, Gerald B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, ISBN 978-0-691-08527-2.
Fréchet, Maurice (1907), "Sur les ensembles de fonctions et les opérations linéaires", C. R. Acad. Sci. Paris, 144: 1414–1416.
Fréchet, Maurice (1904), "Sur les opérations linéaires", Transactions of the American Mathematical Society, 5 (4): 493–499, doi:10.2307/1986278, JSTOR 1986278.
Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, World Scientific, ISBN 978-981-238-043-2.
Grattan-Guinness, Ivor (2000), The search for mathematical roots, 1870–1940, Princeton Paperbacks, Princeton University Press, ISBN 978-0-691-05858-0, MR 1807717.
Halmos, Paul (1957), Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Pub. Co
Halmos, Paul (1982), A Hilbert Space Problem Book, Springer-Verlag, ISBN 978-0-387-90685-0.
Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis, New York: Springer-Verlag.
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.
Holevo, Alexander S. (2001), Statistical Structure of Quantum Theory, Lecture Notes in Physics, Springer, ISBN 3-540-42082-7, OCLC 318268606.
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.
Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, 15, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0819-1, MR 1468229.
Kadison, Richard V.; Ringrose, John R. (1983), Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, New York: Academic Press, Inc.

Kakutani, Shizuo (1939), "Some characterizations of Euclidean space", Japanese Journal of Mathematics, 16: 93–97, doi:10.4099/jjm1924.16.0_93, MR 0000895.
Kline, Morris (1972), Mathematical thought from ancient to modern times, Volume 3 (ed. 3rd), Oxford University Press (diterbitkan 1990), ISBN 978-0-19-506137-6.
Kolmogorov, Andrey; Fomin, Sergei V. (1970), Introductory Real Analysis (ed. Revised English edition, trans. by Richard A. Silverman (1975)), Dover Press, ISBN 978-0-486-61226-3.
Krantz, Steven G. (2002), Function Theory of Several Complex Variables, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2724-6.
Lanczos, Cornelius (1988), Applied analysis (ed. Reprint of 1956 Prentice-Hall), Dover Publications, ISBN 978-0-486-65656-4.
Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars.
Levitan, B.M. (2001), "Hilbert space", dalam Hazewinkel, Michiel (penyunting), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
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.
Marsden, Jerrold E. (1974), Elementary classical analysis, W. H. Freeman and Co., MR 0357693.
Murphy, Gerald J. (1990), C*-algebras and Operator Theory, Academic Press, ISBN 0-12-511360-9.
von Neumann, John (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische Annalen, 102: 49–131, doi:10.1007/BF01782338, S2CID 121249803.
Templat:Narici Beckenstein Topological Vector Spaces
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, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, diterjemahkan oleh Beyer, Robert T., Princeton University Press (diterbitkan 1996), ISBN 978-0-691-02893-4, MR 1435976.
Nielsen, Michael A.; Chuang, Isaac L. (2000), Quantum Computation and Quantum Information (ed. 1st), Cambridge: Cambridge University Press, ISBN 978-0-521-63503-5, OCLC 634735192.
O'Connor, John J.; Robertson, Edmund F. (1996), "Abstract linear spaces", arkib MacTutor History of Mathematics, Universiti St AndrewsCS1 maint: ref=harv (link).
Peres, Asher (1993), Quantum Theory: Concepts and Methods, Kluwer, ISBN 0-7923-2549-4, OCLC 28854083
Prugovečki, Eduard (1981), Quantum mechanics in Hilbert space (ed. 2nd), Dover (diterbitkan 2006), ISBN 978-0-486-45327-9.
Reed, Michael; Simon, Barry (1980), Functional Analysis, Methods of Modern Mathematical Physics, Academic Press, ISBN 978-0-12-585050-6.
Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics, Academic Press, ISBN 9780125850025.
Rieffel, Eleanor G.; Polak, Wolfgang H. (2011-03-04), Quantum Computing: A Gentle Introduction (dalam bahasa Inggeris), MIT Press, ISBN 978-0-262-01506-6.
Riesz, Frigyes (1907), "Sur une espèce de Géométrie analytique des systèmes de fonctions sommables", C. R. Acad. Sci. Paris, 144: 1409–1411.
Riesz, Frigyes (1934), "Zur Theorie des Hilbertschen Raumes", Acta Sci. Math. Szeged, 7: 34–38.
Riesz, Frigyes; Sz.-Nagy, Béla (1990), Functional analysis, Dover, ISBN 978-0-486-66289-3.
Templat:Rudin Walter Functional Analysis
Rudin, Walter (1987), Real and Complex Analysis, McGraw-Hill, ISBN 978-0-07-100276-9.
Saks, Stanisław (2005), Theory of the integral (ed. 2nd Dover), Dover, ISBN 978-0-486-44648-6; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.
Templat:Schaefer Wolff Topological Vector Spaces
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, 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.
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.
Stewart, James (2006), Calculus: Concepts and Contexts (ed. 3rd), Thomson/Brooks/Cole.
Stein, E (1970), Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, ISBN 978-0-691-08079-6.
Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
Streater, Ray; Wightman, Arthur (1964), PCT, Spin and Statistics and All That, W. A. Benjamin, Inc.
Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 978-0-8218-4660-5..
Titchmarsh, Edward Charles (1946), Eigenfunction expansions, part 1, Oxford University: Clarendon Press.
Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press.
Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90894-6.
Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, 68, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90427-6, MR 0566954.
Weyl, Hermann (1931), The Theory of Groups and Quantum Mechanics (ed. English 1950), Dover Press, ISBN 978-0-486-60269-1.
Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33071-8, Zbl 0645.46024.

Pautan luar

Hazewinkel, Michiel, penyunting (2001), "Hilbert space", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Hilbert space at Mathworld
245B, notes 5: Hilbert spaces by Terence Tao