Hilbert space

For the Hilbert space-filling curve, see Hilbert curve.

Hilbert spaces can be used to study the harmonics of vibrating strings.

The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It takes the mathematics used in two and three dimensions, and asks what happens if there are more than three dimensions. It is named after David Hilbert.

It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer). Hilbert spaces are used in ergodic theory which forms the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term "Hilbert space" for the abstract concept underlying many of these diverse applications. The success of Hilbert space methods started a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

References

• 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 0821800493.
• Bourbaki, Nicolas (1986), Spectral theories, Elements of mathematics, Berlin: Springer-Verlag, ISBN 0201007673.
• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3540136279.
• Boyer, Carl Benjamin; Merzbach, Uta C (1991), A History of Mathematics (2nd ed.), John Wiley & Sons, Inc., ISBN 0-471-54397-7.
• Brenner, S.; Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods (2nd ed.), Springer, ISBN 0-3879-5451-1.
• 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 (Reprint of Wadsworth and Brooks/Cole 1992 ed.), American Mathematical Society Bookstore, ISBN 0821847902, http://books.google.com/books?as_isbn=0821847902.
• Folland, Gerald B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, ISBN 0-691-08527-7.
• Fréchet, Maurice (1907), "Sur les ensembles de fonctions et les opérations linéaires", C. R. Acad. Sci. Paris 144: 1414–1416.
• Fréchet, Maurice (1904–1907), Sur les opérations linéaires.
• Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, World Scientific, ISBN 981-238-043-4.
• 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 0387906851.
• Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis, 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, http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D27779.
• 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.
• Kakutani, Shizuo (1939), "Some characterizations of Euclidean space", Jap. J. Math. 16: 93–97, MR 0000895.
• Kline, Morris (1972), Mathematical thought from ancient to modern times, Volume 3 (3rd ed.), Oxford University Press (published 1990), ISBN 978-0195061376.
• Kolmogorov, Andrey; Fomin, Sergei V. (1970), Introductory Real Analysis (Revised English edition, trans. by Richard A. Silverman (1975) ed.), Dover Press, ISBN 0-486-61226-0.
• 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 (Reprint of 1956 Prentice-Hall ed.), Dover Publications, ISBN 048665656X, http://books.google.com/books?as_isbn=048665656X.
• 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.
• O'Connor, John J; Edmund F. Robertson (1996). "Abstract linear spaces". MacTutor History of Mathematics archive.  .
• Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars, http://books.google.com/?id=VfUKAAAAYAAJ&dq=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1#v=onepage&q=.
• B.M. Levitan (2001), "Hilbert space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=H/h047380.
• Marsden, Jerrold E. (1974), Elementary classical analysis, W. H. Freeman and Co., MR 0357693.
• Prugovečki, Eduard (1981), Quantum mechanics in Hilbert space (2nd ed.), Dover (published 2006), ISBN 978-0486453279.
• Reed, Michael; Simon, Barry (1980), Functional Analysis, Methods of Modern Mathematical Physics, Academic Press, ISBN 0-12-585050-6.
• Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics, Academic Press, ISBN 0-12-5850002-6.
• 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 0-486-66289-6.
• Rudin, Walter (1973), Functional analysis, Tata MacGraw-Hill.
• Rudin, Walter (1987), Real and Complex Analysis, McGraw-Hill, ISBN 0-07-100276-6.
• Saks, Stanisław (2005), Theory of the integral (2nd Dover ed.), Dover, ISBN 978-0486446486; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.
• Schmidt, Erhard (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo 25: 63–77, doi:10.1007/BF03029116.
• 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 883081.
• Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 9789810223861, MR 1626401.
• Stewart, James (2006), Calculus: Concepts and Contexts (3rd ed.), Thomson/Brooks/Cole.
• Stein, E (1970), Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, ISBN 0-691-08079-8.
• 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.
• Titchmarsh, Edward Charles (1946), Eigenfunction expansions, part 1, Oxford University: Clarendon Press.
• Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press.
• von Neumann, John (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische Annalen 102: 49–131, doi:10.1007/BF01782338.
• 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, Princeton University Press (published 1996), ISBN 978-0-691-02893-4, MR 1435976.
• 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 566954.
• Weyl, Hermann (1931), The Theory of Groups and Quantum Mechanics (English 1950 ed.), Dover Press, ISBN 0-486-60269-9.
• Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 0-521-33071-8, Zbl 0645.46024.

Other websites

• Hilbert space at Mathworld
• 245B, notes 5: Hilbert spaces by Terence Tao