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# Topic: Riesz representation theorem

###### In the News (Fri 7 Mar 14)

 PlanetMath: Riesz representation theorem (of linear functionals on function spaces) This entry should not be mistaken with the entry on the Riesz representation theorem of bounded linear functionals on an Hilbert space. The Riesz representation theorem(s) provided here basically state that linear functionals on certain spaces of functions can be seen as integration against measures. This is version 4 of Riesz representation theorem (of linear functionals on function spaces), born on 2007-08-12, modified 2007-08-12. planetmath.org /encyclopedia/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces.html   (291 words)

 Riesz representation theorem - Wikipedia, the free encyclopedia 3 The representation theorem for the dual of C0(X) This theorem establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. Gray, The shaping of the Riesz representation theorem: A chapter in the history of analysis, Archive for History in the Exact Sciences, Vol 31(3) 1984-85, 127-187. en.wikipedia.org /wiki/Riesz_representation_theorem   (839 words)

 Riesz representation theorem - Definition, explanation There are several well-known theorems in functional analysis known as the Riesz representation theorem. This theorem establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics. www.calsky.com /lexikon/en/txt/r/ri/riesz_representation_theorem.php   (703 words)

 Riesz representation theorem   (Site not responding. Last check: 2007-10-09) Theorem 1 (Riesz Representation Theorem) Let H be a Hilbert space and... Theorem (Riesz Representation Theorem) Let X be a locally com-... A Riesz representation theorem for cone-valued functions, Walter Roth... www.scienceoxygen.com /math/406.html   (159 words)

 Bra-ket notation - Wikipedia, the free encyclopedia In quantum mechanics, the state of a physical system is identified with a vector in a complex Hilbert space, H. The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically isomorphic. en.wikipedia.org /wiki/Bra-ket_notation   (919 words)

 Frigyes Riesz   (Site not responding. Last check: 2007-10-09) In 1945 Riesz was appointed to the chair of mathematics at the University of Budapest. Riesz was a co-founder of functional analysis, his work of 1910 marks the start of operator theory. Such fundamental results as the Riesz Representation theorem (the form of functionals on the space of continuous functions), the Riesz-Fischer theorem (completeness of L^2 spaces and identification of different Hilbert spaces) bear his name. www.math.u-szeged.hu /confer/fejerriesz/Riesz.htm   (365 words)

 PlanetMath: proof of Riesz representation theorem for separable Hilbert spaces "proof of Riesz representation theorem for separable Hilbert spaces" is owned by rspuzio. This is version 2 of proof of Riesz representation theorem for separable Hilbert spaces, born on 2004-09-03, modified 2005-01-25. I wanted to add a proof for the general case, but you might like to change your entry to have only one proof for the theorem. www.planetmath.org /encyclopedia/ProofOfRieszRepresentationTheoremForSeparableHilbertSpaces.html   (160 words)

 Course Notes A Riesz space is an ordered Q-vector space, which is a lattice (it is then automatically distributive). Let X be a compact Hausdorff, C(X) the Riesz space of continuous function on X. One basic remark is that the Riesz space B(X) of bounded Baire functions on X is the sigma completion of C(X). In this way both the Riesz-Markov representation theorem and the monotone convergence theorem get a natural interpretation: integral of Baire functions (and measure of Baire subsets) can be directly defined by this universal property of B(X), using the next point. www.md.chalmers.se /~coquand/measure.html   (265 words)

 Hilbert space - Wikipedia, the free encyclopedia In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H into the base field), which is itself a Hilbert space. Indeed, the Riesz representation theorem states that to every element φ of the dual H' there exists one and only one u in H such that If a linear operator has a closed graph and is defined on all of a Hilbert space, then, by the closed graph theorem in Banach space theory, it is necessarily bounded. en.wikipedia.org /wiki/Hilbert_space   (1887 words)

 About The Book Theorem 16.6 in Chapter IV contains an interesting proof that (1+1/n)ⁿ is increasing and convergent. Included is a discussion of Cesàro summability (Section 77) and Hardy's Tauberian theorem (Theorem 79.1) with an application to Fourier series (Theorem 79.3). The appendix summarizes some of the key definitions and theorems concerning vector spaces that are used in the book. condor.depaul.edu /~rjohnson/foma/about.html   (757 words)

 [No title] I pause a little, and then come up with the Central Limit Theorem, which then Mather asks to formulate precisely and to explain in detail where the weak convergence of measures comes in. Deduce from it the fundamental theorem of algebra. Now a question on the Liouville theorem: (here comes a beautiful question) Let F, G be meromorphic functions such that F^n+G^n=1. www.math.princeton.edu /graduate/generals/bufetov_alexander   (921 words)

 FuncAna The Riesz representation theorem for linear functionals on a Hilbert space. Converse of the spectral theorem: every operator that admits the spectral decomposition with real eigenvalues is compact and self-adjoint. Reduction to an integral equation and proof of the spectral theorem. www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html   (947 words)

 [No title] If K is a compact Hausdorff space then the dual of C(K) is the space of regular Borel complex measures on K; this is what was called the Riesz representation theorem when I was young. There are lots of theorems that have the name Riesz attached to them, and several known as the "Riesz Representation Theorem", but the particular one you're asking about in (b) says that every bounded linear functional on a Hilbert space is given by the inner product with a vector in the Hilbert space. Frigyes Riesz was the first to prove this for general Hilbert spaces (without assuming separability): F. Riesz, "Zur Theorie des Hilbertschen Raumes", Acta Sci. www.math.niu.edu /~rusin/known-math/98/riesz   (380 words)

 Department of TFPT   (Site not responding. Last check: 2007-10-09) The main properties of harmonic and subharmonic functions (characteristic properties, maximum principle, the mean-values properties, the Riesz representation theorem of a subharmonic function, Jensen's formula), the properties of upper semicontinuous functions are considered in this special course. Harnac's inequality and Harnac's theorem for harmonic functions. Theorem on the existence of the Riesz measure. www.franko.lviv.ua /faculty/mechmat/Departments/TFTJ/Web/subharmeng.htm   (163 words)

 Ernest Schimmerling The name reverse mathematics comes from the fact that, the reverse is also true - if certain axioms prove a theorem, they are logically equivalent to it. Then I will state the Riesz Representation Theorem and prove that it is equivalent to a number of other statements. The main result will be that both the existence of orthogonal projection onto closed subspaces of a Hilbert space and the Riesz Representation Theorem are equivalent to ACA0. www.math.cmu.edu /users/eschimme/seminar/simic.html   (202 words)

 University of Chicago Department of Mathematics Applications to representation theory of compact groups (the Peter-Weyl theorem) and an introduction to the calculus of variations. Hartogs' Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces. Inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem. www.math.uchicago.edu /firstyear.html   (506 words)

 PlanetMath: Riesz representation theorem proof of Riesz representation theorem for separable Hilbert spaces This is version 6 of Riesz representation theorem, born on 2004-02-16, modified 2004-02-17. Isn't there another version of this theorem which states that planetmath.org /encyclopedia/RieszRepresentationTheorem.html   (101 words)

 Real and Complex Analysis : McGraw-Hill Professional Books   (Site not responding. Last check: 2007-10-09) This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. books.mcgraw-hill.com /getbook.php?isbn=0070542341   (224 words)

 Wash U Graduate Program : Graduate Course Offerings Inverse function theorem, implicit function theorem and theorem on rank. Prerequisites: We shall refer to theorems from advanced calculus at the level of Math 411 and Math 412, and linear algebra at the level of Math 309. This course covers complex tori and their line bundles, Appel-Humbert Theorem, Theta functions and Theta groups, addition formula and multiplication formula for Theta functions, embeddings of abelian varieties into projective spaces via Theta functions, and moduli spaces of abelian varieties. www.math.wustl.edu /academics/graduate/courses.html   (2228 words)

 Riesz representation theorem   (Site not responding. Last check: 2007-10-09) The Riesz representation theorem in functional analysis establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics. The Riesz representation theorem states that every element of H ' can be written in this form, and that furthermore the assignment Φ(x) = φ www.findword.org /ri/riesz-representation-theorem.html   (601 words)

 [No title]   (Site not responding. Last check: 2007-10-09) It is expected that the student has mastered the basic principles of analysis of real functions of one and several variables. convergence theorems for the Lebesgue integral: Fatou's Lemma, monotone convergence theorem, Lebesgue dominated convergence theorem. Chapters 1-6 (from Chapter 2 omit the Riesz representation theorem and from Chapter 5 an abstract approach to the Poisson Integral), Chapter 8 (omit convolutions and distribution functions). www.lehigh.edu /~math/realana.html   (290 words)

 Measure and Integral   (Site not responding. Last check: 2007-10-09) From the topics on measure and integration on R^n, the covering theorems, differentiation of measures, density topology and approximately continuous functions are included. New proofs of the Rademacher and Besicovitch theorems are presented. A part of the book is devoted to the study of the theory of distributions, Fourier transform, approximation in function spaces and degree theory. www.karlin.mff.cuni.cz /lat/~lukes/mai.htm   (208 words)

 Descriptions of spring 2005 courses in the Rutgers-New Brunswick Math Graduate Program Topics to be covered: the Riesz representation theorem for positive linear functionals on C(X), the Birkhoff ergodic theorem, the Marcinkiewicz interpolation theorem, the Riesz-Thorin interpolation theorem, the Fourier transform and elementary theory of singular integrals. We outline the proofs of these results, including the positive mass theorem of Schoen and Yau on which the proof of the last case relies. Representation theory studies the ways a given group or algebra can be represented as a group (or algebra) of matrices. www.math.rutgers.edu /grad/courses/spring_2005_descriptions.html   (3510 words)

 Numerical Notes: Math Archives A definition of a tensor which is based on a particular concrete representation artifically complicates manipulations. That is, we have a basis for the space of linear operators on V; and where we have a basis, we have a concrete representation. The Riesz representation theorem actually holds for any separable Hilbert space, even the infinite-dimensional ones. www.cs.berkeley.edu /~dbindel/blog/archives/cat_5   (1882 words)

 Graduate Course Descriptions Lebesgue integration, measure theory, convergence theorems, the Riesz representation theorem, Fubini’s theorem, complex measures. According to it, there are topological restrictions on the way the Riemann surface of a function representable by radicals covers the complex plane. The following topics are prerequisites for the course: minimal knowledge of finite dimensional Representation Theory for a semi-simple group G (over complex numbers), singular homology and cohomology, basic Morse theory, a few definitions (like affine and projective varieties, line bundles etc.) from basic Algebraic Geometry. www.math.toronto.edu /graduate/courses/descriptions.html   (4148 words)

 Math603-00F   (Site not responding. Last check: 2007-10-09) Calculus of one and several variables, including line integrals, surface integrals, Stokes' theorem, the Implicit and Inverse Function Theorems, pointwise and uniform convergence of sequences of functions, integration and differentiation of sequences, the Weierstrass Approximation Theorem, the existence and uniqueness of solutions of ordinary differential equations. Dual-spaces and their conjugates, the Riesz-Fisher Theorem, the Riesz Representation Theorem for bounded linear functionals on C(X), the Riesz Representation Theorem for C(X), the Hahn-Banach Theorem, the Closed Graph and Open Mapping Theorems, the Principle of Uniform Boundedness, Alaoglu's Theorem, Hilbert spaces, orthogonal systems, Fourier series, Bessel's inequality, Parseval's formula, convolutions, Fourier transform, distributions. Marcinkiewicz interpolation theorem, Calderon-Zygmund decomposition lemma, singular integrals and L^p estimates for Newtonian potential. www.nd.edu /~b1hu/math603-00F/math603.html   (306 words)

 Re: Radon-Nikodym and Riesz Representation theorems Re: Radon-Nikodym and Riesz Representation theorems, A N Niel Re: Radon-Nikodym and Riesz Representation theorems, The World Wide Wade Re: Radon-Nikodym and Riesz Representation theorems, Herman Rubin www.usenet.com /newsgroups/sci.math/msg11407.html   (176 words)

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