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# Topic: Hilbert space

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 Encyclopedia: Hilbert space   (Site not responding. Last check: 2007-10-22) Hilbert spaces provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics. Hilbert space is not a space of simple points, rather it is a space of functions at a higher level of mathematical abstraction. www.nationmaster.com /encyclopedia/Hilbert-space   (680 words)

 Hilbert Space The angular momentum of a particle relative to a direction in space is incompatible with the angle of rotation of that particle in a plane perpendicular to that direction in space. If physical space really is a continuum as in Newton's and Einstein's classical physics, then there is a nondenumerable or noncountable infinity of dimensions to the quantum Hilbert space for the motion of this single particle. Although the total quantum state is a ray in which only the direction in Hilbert space is physically observable, the projective coordinates of this ray, onto a frame of reference of basic rays, are two-real dimensional vectors in the complex plane in which both the magnitude and relative direction are important. www.qedcorp.com /pcr/pcr/hilberts.html   (2670 words)

 Rigged Hilbert space - Wikipedia, the free encyclopedia In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. A rigged Hilbert space is a pair (H,Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous. en.wikipedia.org /wiki/Rigged_Hilbert_space   (405 words)

 natural religion > glossary > Hilbert space   (Site not responding. Last check: 2007-10-22) Hilbert space, named for David Hilbert, is a mathematical construction derived from ordinary three dimensional Euclidean space. Since each point in Hilbert space may be used to represent an infinite series, we may think of Hilbert space as a space of functions, or function space. Hilbert space is a metric space, which means that we can define a distance between two functions. www.naturaltheology.net /Glossary/hilbertSpace.html   (430 words)

 Hilbert Hilbert was a member of staff at Königsberg from 1886 to 1895, being a Privatdozent until 1892, then as Extraordinary Professor for one year before being appointed a full professor in 1893. Hilbert's eminent position in the world of mathematics after 1900 meant that other institutions would have liked to tempt him to leave Göttingen and, in 1902, the University of Berlin offered Hilbert Fuchs' chair. Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hilbert.html   (1587 words)

 PlanetMath: Hilbert space In particular, a Hilbert space is a Banach space in the norm induced by the inner product, since the norm and the inner product both induce the same metric. Any finite-dimensional inner product space is a Hilbert space, but it is worth mentioning that some authors require the space to be infinite dimensional for it to be called a Hilbert space. This is version 8 of Hilbert space, born on 2002-02-13, modified 2004-12-07. planetmath.org /encyclopedia/HilbertSpace.html   (168 words)

 natural theology > development > 3 physics > 7 Hilbert space   (Site not responding. Last check: 2007-10-22) A phase space is some sort of mathematical transformation of physical space which shines a spotlight on a particular aspect of physical process. The amount of information carried by a point in phase space depends on the number of components in the vector that represents it, which in turn depends upon the number of dimensions in the phase space. Jacket: 'Although the concept of space is of fundamental importance in both physics and philosophy, until the publication of this book, the idea of space had never been treated in terms of its historical development. www.naturaltheology.net /Development/Dev03_Physics/phys07HilbertSpace.html   (2149 words)

 Operators on Hilbert Space A Hilbert space theory of differential operators is complicated by the fact that differentiability and square integrability are really two quite different concepts. Hilbert space theory and differential equation theory can be used in harmony, without the necessity to give up such things as continuum wave functions because of their nonnormalizability. At the beginning of this century the theory of differential operators was already rather extensively developed, especially with regard to the classification of the singularities which could occur in the solulions as a consequence of singularities in the coefficients, or of the infinitude of the interval of integration. delta.cs.cinvestav.mx /~mcintosh/comun/quant/node3.html   (1649 words)

 PlanetMath: Hilbert parallelotope The Hilbert parallelotope enjoys a remarkable universality property -- every second countable metric space is homeomorphic to a subset of the Hilbert parallelotope. Cross-references: second countability is hereditary, metric space, second countable, property, compact subset, compact, Tychonoff's theorem, closed interval, countably infinite, homeomorphic, topological space, scalars, field, Hilbert space, subset, closed This is version 2 of Hilbert parallelotope, born on 2004-09-24, modified 2005-04-17. planetmath.org /encyclopedia/HilbertCube.html   (167 words)

 Basic Structures: Hilbert space Hilbert spaces are mentioned in most textbooks on quantum mechanics and functional analysis [3]. Of course, all linear operators on a finite dimensional space are bounded anyway, and the B is used mostly for conformity with the infinite dimensional case. Abstract and concrete Hilbert spaces are completely analogous to "abstract vectors", and "vectors written out in components" in ordinary analytical geometry: the passage between the two amounts to a choice of basis, and often a computation can be simplified substantially by choosing a basis adapted to the problem. www.imaph.tu-bs.de /qi/lecture/qinf21.html   (1350 words)

 math lessons - Hilbert space Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. In quantum mechanics, several interesting unbounded operators are defined on a dense subspace of Hilbert space. www.mathdaily.com /lessons/Hilbert_space   (1716 words)

 Hilbert Space Explorer Home Page   (Site not responding. Last check: 2007-10-22) Hilbert space [external] underlies the foundation of quantum mechanics, and there is a strong physical and philosophical motivation to understand its properties. A practical problem with this approach is that the direct definition of a Hilbert space is large and awkward to work with, unless we already have available an underlying theory of vector spaces, etc. (which at this point I do not have). The set of closed subspaces of Hilbert space obey the laws of a simple equational algebra called "orthomodular lattice algebra." This algebra is sometimes called "quantum logic," and it can be used as the basis for a propositional calculus that resembles but is somewhat weaker than standard (classical) propositional calculus. us.metamath.org /mpegif/mmhil.html   (2104 words)

 Topological Space   (Site not responding. Last check: 2007-10-22) Such a space, that is one with closure and a metric, may be of any dimension. These spaces are usually defined by their components and we usually call them vector spaces. By completeness (see the Math Appendix) we mean that, in the linear vector space, any function in the space may be expressed as a linear combination of the functions making up the space. www.chem.brown.edu /chem277/space.html   (566 words)

 The Hilbert Space Filling Curve   (Site not responding. Last check: 2007-10-22) The Hilbert space filling curve is a one dimensional curve which visits every point within a two dimensional space. The actual Hilbert Curve is not a member of this family, it is the limit that the sequence approaches. This apparent riddle is solved as although none of the family of curves leading to the Hilbert curve crosses itself the final curve does cross itself all over the place. www.dcs.napier.ac.uk /~andrew/hilbert.html   (363 words)

 [No title] A complete space is one in which all Cauchy sequences have a limit.) One particular example of a Hilbert space is the space of all functions whose square has a finite integral (in the sense of Lebesgue). Another example is the space of those functions, all of whose derivatives of order up through \$r\$, belong to \$L_2\$. First of all, they provide us with Hilbert spaces that are useful for presenting the weak/variational forms of elliptic problems. www.math.niu.edu /~rusin/known-math/00_incoming/sobolev   (816 words)

 Hilbert Space A hilbert space (biography) is a banach space with a dot product. Thus far, a banach space could have any norm, but in a hilbert space, the norm is tied to the dot product. If s is a complex vector space it is certainly a real vector space, and the distance metric is unchanged, thus s becomes a real vector space that is also a complete metric space, i.e. www.mathreference.com /top-ban,hilbert.html   (1106 words)

 Spaces like Hilbert space (from topology) --  Encyclopædia Britannica   (Site not responding. Last check: 2007-10-22) More results on "Spaces like Hilbert space (from topology)" when you join. Every topological space consists of: (1) a set of points; (2) a class of subsets defined axiomatically as open sets; and (3) the set operations of union and intersection. To emphasize the importance of keeping undefined mathematical terms totally abstract he once said, “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” His work with integral equations in 1909 led later in... www.britannica.com /eb/article-69115   (747 words)

 [No title] We use Hilbert module language to study the semi-invariant subspaces of a family of weighted Fock spaces and their quotients that includes the Full Fock space, the symmetric Fock space, the Direichet algebra, and the reproducing kernel Hilbert spaces with Nevanlinna-Pick kernel. In viewing contraction operators on a Hilbert space as contractive Hilbert modules over the disk algebra, the Sz.-Nagy-Foias canonical model can be seen to correpond to a short exact resolution of the given module by isometric modules. In Banach space theory the 'little Grothendieck inequality' is a fundamental tool and known for its connection to Khintchine's inequality and projection constants. www.math.tamu.edu /research/workshops/linanalysis/speakers.html   (1057 words)

 Hilbert space   (Site not responding. Last check: 2007-10-22) Hilbert's gift helps symphony buy tower: Washington Place to ease space crunch. a village the size of Hilbert, population 1,200. Sobolev spaces, denoted by are another example of Hilbert spaces, which are used very often in the field of Partial differential equations. hallencyclopedia.com /Hilbert_space   (1824 words)

 Space-Filling Curves   (Site not responding. Last check: 2007-10-22) For example, the name Hilbert space-filling curve should properly be used only for the limit curve reached as the level parameter of the Hilbert curves tends to infinity. The Hilbert curve starts and ends at opposite ends of a side of the unit square. The Peano curves use square cells; but, unlike the Hilbert curve, each cell is subdivided into 9 cells at the next recursive level. www.cs.utexas.edu /users/vbb/misc/sfc/Oindex.html   (511 words)

 Hilbert Space Theory Hilbert spaces provide the mathematical foundation for signal processing theory. In this section we attempt to clearly define some key Hilbert space concepts like vectors, norms, inner products, subspaces, orthogonality, orthonormal bases, and projections. The intent is not to bury you in mathematics, but to familiarize you with the terminology, provide intuition, and leave you with a "lookup table" for future reference. cnx.rice.edu /content/m11007/latest   (86 words)

 Hilbert Space-Filling Curves Although the origin of the multidimensional spaces are 0, the coordinates returned for other scalars depend on the width argument. Although the origin of the multidimensional spaces are at (0,..., 0), the coordinates returned for other indexes depend on the width argument. slib/phil-spc.scm is a Scheme implementation of Hilbert conversions between non-negative scalars and non-negative coordinate vectors of arbitrary size. www-swiss.ai.mit.edu /~jaffer/Geometry/HSFC.html   (1080 words)

 A Hilbert Space for Stochastic Processes   (Site not responding. Last check: 2007-10-22) The result of primary concern here is the construction of a Hilbert space for stochastic processes. The space consisting of random variables X having a finite mean-square value is (almost) a Hilbert space with inner product E[XY]. Parametrically, X(t) is therefore regarded as a "curve" in a Hilbert space. cnx.rice.edu /content/m11258/latest   (493 words)

 Constructs in Hilbert Space   (Site not responding. Last check: 2007-10-22) Cyberspace is potentially larger than physical space, since it could contain a universe fractally divisible to infinity. Whether cognispace or Hilbert space is the larger, or which contains the other, or whether they are in fact identical, remains to be seen. You first visualize Hilbert space as the sum of all possible universes, then you visualize the goal you want to achieve (or problem as solved), then you visualize a path from the present circumstances to the goal in Hilbert space, then you focus in on the Next Step. home.earthlink.net /~eldonenew/hilbert.htm   (819 words)

 Characterizing the Separable Hilbert Space Suppose s is a separable hilbert space with an uncountable orthogonal system. It is a basis for the hilbert space, and a basis in the traditional sense. Therefore s is an R vector space, and a normed vector space. www.mathreference.com /top-ban,orsys.html   (1597 words)

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