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# Topic: Dual space

###### In the News (Sat 17 Aug 19)

 Dual space - Wikipedia, the free encyclopedia The use of the dual space in some fashion is thus characteristic of functional analysis. Consequently, dual space is an important concept in the study of functional analysis. The algebraic dual space is defined for all vector spaces. en.wikipedia.org /wiki/Dual_space   (1795 words)

 IDENTICAL DUAL LATTICES... This paper intends to report on research in which generation of dual space lattices and hyperbolic partition surfaces are at the core of the inquiry. This stage, the impression is that the number of the dual and identical lattice pairs and the associated periodic hyperbolic surfaces, partitions that subdivide the entire space into two complementary and identical subspaces, is infinite. The E.P.R. is the smallest space unit (fundamental region) derived from the Euclidean space by means of the symmetry group that acts on this space. members.tripod.com /vismath7/proceedings/korren.htm   (1258 words)

 PlanetMath: dual space   (Site not responding. Last check: 2007-10-16) The notions of duality extend, in part, from vector spaces to modules, especially free modules over commutative rings. A related notion is the duality in projective spaces. This is version 10 of dual space, born on 2002-02-03, modified 2004-09-17. planetmath.org /encyclopedia/DualBasis.html   (241 words)

 Identical Dual Lattices ans Subdivision of Space   (Site not responding. Last check: 2007-10-16) This identification of the 2-manifolds which divide the space into two identical subspaces point out that there is a direct correspondence between the continuous partition surface, the character of the two complimentary subspaces and the two self dual lattices representing them. Currently, the impression is that the number of the dual and identical lattice pairs and the associated periodic hyperbolic surfaces, partitions that subdivide the entire space into two complementary and identical subspaces, is infinite. As of now it is clear that a general comprehensive notation system of space lattices is urgently required, as well as additional elaboration on the subject of "universal parameters" capable of capturing the nature and quality of the various ordered space lattices and related hyperbolic surfaces. www.mi.sanu.ac.yu /vismath/koren/index.html   (1995 words)

 Wikinfo | Lp space spaces are spaces of p-power integrable functions, and corresponding sequence spaces. If S is the set of natural numbers, the space is usually denoted as ellp (ie., without the space indication). The question of whether all Banach spaces have such an embedding was answered negatively by B. Tsirelson's construction of Tsirelson space in 1974. www.wikinfo.org /wiki.php?title=Lp_space   (778 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-16) I thought that a dual space was when you have a vector space V and you have a transformation L that takes V to another vector space U. And a dual map is a function that assigns U* to a field in V*. The dual space of V, called V*, is the space of all linear functionals on V. What are the kernel and image of this linear transformation L? The kernel is the space of all vectors that are mapped by L to the zero vector (0,0,0). mathforum.org /library/drmath/view/63857.html   (1088 words)

 THE “DUAL KEEL” SPACE STATION [1985]   (Site not responding. Last check: 2007-10-16) The new "Dual Keel" Station was largely based on Lockheed and McDonnell-Douglas designs and its structure was much stiffer and hence easier to stabilize. It would offer additional space for external instruments plus a better microgravity environment than the previous gravity gradient-stabilized "Power Tower.” The number of crewmembers was increased to 8 astronauts because scientists were concerned that a crew of six would be too busy with assembly & maintenance tasks and have no time for research. “Dual Keel” Space Station illustration from the 1986 “Pioneering the Space Frontier” report. www.abo.fi /~mlindroo/Station/Slides/sld038.htm   (222 words)

 Space Island Group - Space Vehicle and Launch Systems   (Site not responding. Last check: 2007-10-16) We are developing a new generation of launch vehicles that will replace the current space shuttle and develop the first real manned commercial presence in space. Their early Space Stations designs centered on the shuttle's hollow, orange external fuel tank, known in the industry as the ET. The 2 solid fuel white rocket boosters attached to the External Fuel Tank drop off 2 minutes after launch and parachute into the sea, the 70,000 pound, aluminum ET stays attached to the shuttle, feeding its liquid fuel to the shuttle's engines during the rest of its 8-1/2 minute flight to orbit. www.spaceislandgroup.com /vehicles-systems.html   (574 words)

 Dual Keel Space Station - 1985 It would offer additional space for external instruments plus a better microgravity environment than the previous gravity gradient-stabilised 'Power Tower.' The number of crew members was increased to 8 astronauts because scientists were concerned that a crew of six would be too busy with assembly and maintenance tasks and have no time for research. The Dual Keel design was found to provide poor access for maintenance, visibility from habitable areas was obstructed, there was no provision for internal vehicle 'hands on' access to EVA equipment. The Space Station was briefly suspended by a major row in the summer of 1986 when NASA tried to reassign some habitation module responsibilities from the Johnson Space Center to the Marshall Space Flight Center. www.astronautix.com /craft/duan1985.htm   (1072 words)

 China: Dual - Use Space Technology Over time, new space launch vehicles have been developed whose missile pedigree is far more attenuated, and whose applicability as engines of war is rather difficult to envision. The space launch industry is extremely sensitive to questions of launcher reliability, with launchers exhibiting reliability lower than the prevailing 90%-95% rates facing potentially prohibitive insurance premiums. Unlike space launch vehicles, the difference between 75% reliability and 90% reliability of the Chinese missile force would have no material bearing on the quality of the existential deterrent, in either Chinese or American calculations. www.fas.org /spp/starwars/congress/1998_h/h980625-pike.htm   (4044 words)

 An Introduction to Banach Space Theory Section 1.10 introduces the dual space of a normed space, and has the characterizations up to isometric isomorphism of the duals of direct sums, quotient spaces, and subspaces of normed spaces. Section 1.12, devoted to separability, includes the Banach-Mazur characterization of separable Banach spaces as isomorphs of quotient spaces of \ell_1, and ends with the characterization of separable normed spaces as the normed spaces that are compactly generated so that the stage is set for the introduction of weakly compactly generated normed spaces in Section 2.8. Schauder's theorem relating the compactness of a bounded linear operator to that of its adjoint is presented, as is the characterization of operator compactness in terms of the bounded-weak*-to-norm continuity of the adjoint. www.math.lsa.umich.edu /~meggin/ibst.html   (2875 words)

 Bra space This type of vector space is called a bra space (after Dirac), and its constituent vectors (which are actually functionals of the ket space) are called bra vectors. There is a one to one correspondence between the elements of the ket space and those of the related bra space. It is possible to define a dual bra space for a ket space of nondenumerably infinite dimensions in much the same manner as that described above. farside.ph.utexas.edu /teaching/qm/lectures/node8.html   (567 words)

 Ray Williamson Ray Williamson is Research Professor of International Affairs and Space Policy in the Space Policy Institute, focusing on international issues in environmental security, Earth observation satellite policy, dual-purpose space technologies, and the commercialization of space-related technologies. Williamson is also a faculty member of the International Space University (ISU), Illkirch, France, teaching general space policy, and issues in commercial remote sensing for the ISU Masters of Space Studies and Summer Session programs. In 1990, Dr. Williamson was Senior Fellow in the Space Policy Institute, George Washington University, where he participated in the study, Documents of the Space Age, a documentary history of the U.S. space program. www.space-services.com /williamson.htm   (340 words)

 Design Process as Dual-Space Search The original idea of two spaces of human problem solving was introduced by Simon and Lea (1974) as a space of instances and space of rules. Kulkarni and Simon (1990) extended the idea of dual-space search to the domain of scientific reasoning, where the two spaces in question were the space of hypotheses (generation of ideas) and the space of experiments (testing of ideas). Generally, the design space is ill-defined in the sense that there are no definite criteria for testing whether a proposed solution is successful or not. mlab.uiah.fi /polut/Design/teoria_design_dual_space.html   (2560 words)

 European Officials Skeptical of U.S. Space Motivations BERLIN — European suspicions that the U.S. government’s invitation to join a long-term space exploration program is a ruse meant to guarantee U.S. domination of near-Earth and deep space appear to be spreading beyond the French circles that first gave voice to them. Hartwig Bischoff, deputy head of the Brussels, Belgium-based European Commission’s space policy office, said he feared that the U.S. President’s Commission on Moon, Mars and Beyond would recommend that a global space exploration effort be modeled after the U.S. Air Force’s Joint Strike Fighter (JSF) program. It details a spectrum of space weaponry — including a hypersonic space plane and spacecraft to neutralize other nations’ satellites — designed to assure that no adversary will be able to use space systems against U.S. forces or cripple U.S. space assets. www.space.com /spacenews/archive04/motivatearch_052504.html   (681 words)

 Dual Space theory Dual Space theory aims to cure many of the problems of physics. #7 is Dual Space' explanation of starlight bending as it passes the Sun. Dual Space theory has been developed over the past five years and exists as a 240 page volume heavy www.dualspace.net /default.html   (400 words)

 [No title] Solovay showed that it is consistent with ZF (plus dependent choice) that every subset of a Polish space has the property of Baire (and thus that every linear functional on a Polish vector space has the property of Baire). Brown claimed that the DIMENSION of the dual space of F^(N) is uncountable, not just that the dual space of F^(N)--as a set--is uncountable. Also, what I later proved (and at that later point stated) is that the DIMENSION of the dual space of F^(N) is at least c. www.math.niu.edu /~rusin/known-math/00_incoming/duals   (2065 words)

 Hilbert space and its dual - Advanced Physics Forums Bras, which are continuous bounded linear functions of kets, must live in the continuous dual space. The other operators, say position and momentum and such, is it safe to say that they live in the complement of the continuous dual space, i.e. They\'re linear operators on the space to the space, not to the base field like the bras are of the kets. www.advancedphysics.org /forum/showthread.php?t=709   (196 words)

 Harvard Gazette: Questions remain about China in space Two Chinese astronauts spent five days in space in early October, boosting national pride and the reputation of China's high-tech industry, but leaving experts scratching their heads about China's military intentions. This is leading to the third phase: the launch of a small space station, possibly as early as 2009, followed within a few years by the launch of a larger space station. The program itself, she said, is an employment program as well as a space program, providing work and training for thousands in the Chinese technology sector. www.news.harvard.edu /gazette/2005/11.03/11-space.html   (738 words)

 Dual Vector Space -- from Wolfram MathWorld In the dual of a complex vector space, the linear functions take complex values. Then a basis corresponds to its dual basis only if it is an orthonormal basis, in which case Dual vector spaces can describe many objects in linear algebra. mathworld.wolfram.com /DualVectorSpace.html   (136 words)

 ESA - Human Spaceflight and Exploration - International Space Station - Dual ISS and Venus transit viewed from Slovakia Dual ISS and Venus transit viewed from Slovakia Tomáš Maruška captured images of the dual ISS/Venus transit from Slovakia. He was helped by his wife Gitka, and their 7 year old daughter, Zuzka. www.esa.int /esaHS/SEMBBK3VQUD_iss_1.html   (170 words)

 Every vector space is the dual of some other vector space For a finite dimensional vector space it is correct, and you even write down the proof: V** is naturally isomorphic to V. www.maths.bris.ac.uk/~maxmg Opinons are mine, not those of my employers. It just tells you that given a space V** and another V (so you already know V** is the double dual of a vector space), there is a one-one mapping between all their members. V** is the dual of the vector space V*, i.e. www.physicsforums.com /showthread.php?p=1011191#post1011191   (1247 words)

 Dual spaces   (Site not responding. Last check: 2007-10-16) Given a vector space defined over a field, such as the real numbers, the dual space is usually defined as the set of all bi-linear mappings of the vectors onto the field. This is unnecessarily abstract, so let us do it this way (for finite spaces): Let V be a vector space with basis elements e1,. Let W be another vector space of the same dimension, with basis elements f1,. www.newton.dep.anl.gov /newton/askasci/1995/math/MATH094.HTM   (230 words)

 Beta-dual space - Wikipedia, the free encyclopedia In functional analysis and related areas of mathematics, the beta-dual or β-dual is a certain linear subspace of the algebraic dual of a sequence space. The beta-dual of an FK-space E is a linear subspace of the continuous dual of E. If E is an FK-AK space then the beta dual is linear isomorphic to the continuous dual. en.wikipedia.org /wiki/Beta-dual_space   (112 words)

 UMass Amherst: Department of Computer Science In the Dual space the interaction can be carried out in three layers: interface, service, and system in more intuitive and natural way for users. To establish Dual relations in the interface layer makes the object surface, paper document even the contents becoming the interface; Dual relations in the service layer allow the content based information index and retrieval. In the smart space the Dual relations in the system layer will be able to carry out implicit interaction with users. www.cs.umass.edu /csinfo/colloquia/DEPT/xu.html   (144 words)

 Dual Problem   (Site not responding. Last check: 2007-10-16) to be a space of the same dimension such that each m-dimensional object in the primal space is represented by a (2-m)-dimensional object in the dual space. Each edge e incident to faces f and g of the polytope is represented by a dual edge e An edge e of a polytope is a silhouette edge with respect to a viewpoint p iff the corresponding dual edge e www.cgl.uwaterloo.ca /~vma/silhouette/dual_problem.html   (102 words)

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