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Topic: Weak topology

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	Weak operator topology - Wikipedia, the free encyclopedia
		In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the functional sending an operator T to the complex number
		The WOT is weaker than the strong operator topology and weaker than the norm topology.
		The linear functionals on the set of bounded operators on a Hilbert space which are continuous in the strong operator topology are precisely those which are continuous in the WOT.
	en.wikipedia.org /wiki/Weak_operator_topology (182 words)

	Weak topology: Definition and Links by Encyclopedian.com - All about Weak topology (Site not responding. Last check: 2007-10-07)
		In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest) topology on the set which makes all the functions continuous.
		A particularly important example of a weak topology is that on a normed vector space with respect to its (continuous) dual.
		The weak topology on X is the weakest topology (the topology with the least open sets) such that all elements of X ' remain continuous.
	www.encyclopedian.com /we/Weak-topology.html (426 words)

	Ultraweak topology - Wikipedia, the free encyclopedia
		In functional analysis, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B
		For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies.
		In general the ultraweak topology is better than the weak operator topology, but is more complicated to define so people usually use the weak operator topology if they can get away with it.
	www.wikipedia.org /wiki/Weak-star_operator_topology (265 words)

	Encyclopedia: Weak topology (Site not responding. Last check: 2007-10-07)
		The term is most commonly used for the initial topology of a normed vector space with respect to its (continuous) dual.
		In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure.
		General topology In functional analysis, the weak-* topology on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual of B(H), the trace class operators on H. See also Topologies on the set of operators on a Hilbert space Categories: Mathematics stubs...
	www.nationmaster.com /encyclopedia/Weak-topology (802 words)

	Weak topology -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		Every normed vector space X is, by using the norm to measure distances, a (A set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality) metric space and hence a topological space.
		If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a (Click link for more info and facts about locally convex topological vector space) locally convex topological vector space.
		Furthermore, the unit ball of X is compact in the weak topology if and only if X is (A personal pronoun compounded with -self to show the agent's action affects the agent) reflexive.
	www.absoluteastronomy.com /encyclopedia/W/We/Weak_topology.htm (478 words)

	Topology glossary - Wikipedia, the free encyclopedia
		The topology T is the smallest topology on X containing B and is said to be generated by B.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
	en.wikipedia.org /wiki/Topology_Glossary (4512 words)

	Muscle Pain Weakness -- Recommendations and Resources (Site not responding. Last check: 2007-10-07)
		A weak acid is an acid that does not fully ionize in solution; that is, if the acid was represented by the general formula AH, then in aqueous solution a significant amount of undissolved AH still remains.
		Weak references may be used to solve the problem of circular references if the reference cycles are broken by replacing strong references with weak references.
		Weak references are also used to minimize the number of unnecessary objects in memory by allowing the program to indicate which objects are not critical by only weakly referencing them.
	www.becomingapediatrician.com /health/101/muscle-pain-weakness.html (761 words)

	star topology (Site not responding. Last check: 2007-10-07)
		Topology - Star (1 of 2) With a star topology, the workstations, fileservers, printers etc. are attached via drop cables to a central hub or multiport repeater.
		With the star topology, all cables run from the computers to a central location, where they are connected to a device called a hub.
		A star topology consists of a backbone (main circuit) to which a number of outgoing lines can be attached ("dropped"), each providing one or more connection port for device to attach to.
	www.maps-universe.com /articles/30/star-topology.html (553 words)

	PlanetMath: weak-* topology (Site not responding. Last check: 2007-10-07)
		as a subbase for the topology (that is finite intersections of such sets form the basis).
		topology is the Alaoglu's theorem which asserts that for
		This is version 5 of weak-* topology, born on 2005-03-08, modified 2005-03-09.
	planetmath.org /encyclopedia/WeakTopology.html (188 words)

	An Introduction to Banach Space Theory
		Section 2.4 develops the properties of topologies induced by families of functions, with special emphasis on the topology induced on a vector space X by a subspace of the vector space of all linear functionals on X.
		The study of the weak topology of a normed space begins in earnest in Section 2.5.
		Gantmacher's theorem is obtained, as well as the equivalence of the weak compactness of a bounded linear operator to the weak*-to-weak continuity of its adjoint.
	www.math.lsa.umich.edu /~meggin/ibst.html (2875 words)

	Weak topology - Definition up Erdmond.Com
		An example of a weak topology that is particularly important in functional_analysis is that on a normed_vector_space with respect to its (continuous) dual.
		If ''X'' is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and ''X'' is a locally convex topological_vector_space.
		topology on ''X'' ' by requiring that it be the weakest topology such that for every ''x'' in ''X'', the substitution map :Φ''x'' : ''X'' ' → R or C defined by :Φ''x''(φ) = φ(''x'') remains continuous.
	www.erdmond.com /Weak_topology.html (382 words)

	PlanetMath: initial topology (Site not responding. Last check: 2007-10-07)
		the product topology is initial with respect to the projections and a subspace topology is initial with respect to the embedding.
		The initial topology is sometimes called topology generated by a family of mappings [2], weak topology [4] or projective topology.
		This is version 5 of initial topology, born on 2005-09-10, modified 2005-09-29.
	planetmath.org /encyclopedia/InitialTopology.html (199 words)

	PlanetMath: $\mathcal{C}^r$ topologies (Site not responding. Last check: 2007-10-07)
		Whitney (or strong) topology is a topology assigned to the space
		with the weak or strong topologies is denoted by
		is compact, the weak and strong topologies coincide.
	planetmath.org /encyclopedia/WeakMathcalCrTopology.html (189 words)

	Topology glossary
		Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and on definitions that are fundamental to a broad range of areas.
		A collection of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase.
		If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of all unions of finite intersections of elements of B.
	www.sciencedaily.com /encyclopedia/topology_glossary (3712 words)

	Research
		topologies on the (nonempty) closed (or compact) subsets of a topological space X have been studied from the beginning of the 20th century.
		The interest for weak topologies arises among others from the fact that under some natural conditions they are measurably compatible, which in turn allows to express the multifunction measurability as ordinary measurability of an associated single-valued function.
		The so-called generalized compact-open topology on the space of partial maps with domains that are closed in a topological space X has been studied in connection with problems arising in differential equations, in mathematical economics, in convergence of dynamic programming models and other fields.
	www.uncp.edu /home/laszlo/research.html (1016 words)

	Bounded weak star and general strict topology (Site not responding. Last check: 2007-10-07)
		Suppose that B is a Banach algebra with bounded approximate identity, and that X is a is a left Banach module over B. The strict topology on X is the topology generated by the set {p_b: b in B} of semi-norms, where p_b(x) = bx
		A final result shows that, if G is a locally compact abelian group with Haar measure, then G is compact if and only if the bounded weak star topology induced in L p(G) (1 < p <=infinity) by Lq(G) (1/q +1/p = 1) is the strict topology induced by L^1(G).
		From this it follows that, in its strict topology, the Banach space of bounded analytic functions on the unit disc is a topological algebra under convolution.
	www.mth.msu.edu /~shapiro/Pubvit/Downloads/BWT_GST/BWST_GST.html (190 words)

	Product Space
		The topology of p is defined by a base, where a base set in p is given by the cross product of open sets in the component spaces.
		In the weak topology, each base set in p is built from component base sets as before, but only finitely many of these base sets are used.
		Once again, verify that the base for the weak topology is valid, and the topology of p includes all finite cross products of open sets in the component spaces, where the remaining components are unconstrained.
	www.mathreference.com /top,prod.html (1765 words)

	Weak operator topology -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		The WOT is weaker than the (Click link for more info and facts about strong operator topology) strong operator topology and weaker than the (Click link for more info and facts about norm topology) norm topology.
		The (Click link for more info and facts about weak-star topology) weak-star topology is stronger than the WOT.
		The (Click link for more info and facts about linear functional) linear functionals on the set of bounded operators on a Hilbert space which are continuous in the (Click link for more info and facts about strong operator topology) strong operator topology are precisely those which are continuous in the WOT.
	www.absoluteastronomy.com /encyclopedia/w/we/weak_operator_topology.htm (222 words)

	PlanetMath: weak-* topology of the space of Radon measures (Site not responding. Last check: 2007-10-07)
		PlanetMath: weak-* topology of the space of Radon measures
		"weak-* topology of the space of Radon measures" is owned by stevecheng.
		This is version 1 of weak-* topology of the space of Radon measures, born on 2005-07-07.
	planetmath.org /encyclopedia/WeakTopologyOfTheSpaceOfRadonMeasures.html (230 words)

	R. Lowen (Site not responding. Last check: 2007-10-07)
		Topology Atlas Conference Abstracts Document # caai-68.htm
		Considering these structures on E and E' we are able to extend important fundamental results in functional analysis to "approximation versions" of these results.
		Thus we are able to prove new characterizations of the dimension, completeness and reflexivity of normed spaces, we obtain Tschebyscheff-type approximation formulas between the various types of convergences on E, E' and E", and making use of the measure of compactness of the unit ball we obtain a 0-1 law for reflexivity.
	www.utm.edu /staff/jschomme/topology/c/a/a/i/68.htm (84 words)

	A Hilbert space (Site not responding. Last check: 2007-10-07)
		(iii) the operator norm is not continuous with respect to the strong operator topology and the weak operator topology;
		(v) the operation multiplication is continuous in neither weak nor strong operator topology.
		Hence (i) implies that the weak operator topology and the strong operator topology don't coincide on B(H).
	web.um.ac.ir /~moslehian/cfa/Ot15.htm (309 words)

	:: weak :: related - ( ankle apologize nature my writing cervix entity ...
		weak ending an unstressed syllable in a normally stressed place at the end of a verse-line.
		weak moment a time when one is unusually compliant or temptable.
		i apologize for the weak nature of my writing
	www.spell-dictionary.com /db/weak (199 words)

	Untitled Document
		The topology on the state space induced in this way is the so called weak*-topology.
		It is natural to wonder if for a unital C-algebra a given generalized Dirac operator on it will induce a metric topology* on the state space equal to the weak*-topology.
		The main purpose of this thesis is to explore the possibility of constructing Dirac operators which will induce the weak-topology on the state space of a certain C-algebra.
	www.math.ku.dk /cal/events/1565.htm (335 words)

	Weak Topology Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-07)
		Looking For weak topology - Find weak topology and more at Lycos Search.
		Find weak topology - Your relevant result is a click away!
		Look for weak topology - Find weak topology at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Weak_topology (514 words)

	Seminar on 3/13/2002 (Site not responding. Last check: 2007-10-07)
		Most probably the reason for studying weak* sequential closures by S.Banach and S.Mazurkiewicz was the lack of acquaintance of S.Banach and his school with concepts of general topology.
		Although the name "General topology" was introduced later, the subject did already existed, F. Hausdorff (1914) introduced topological spaces, P.Alexandroff-P.Urysohn (1924) introduced compact spaces and A.Tychonoff (1929) proved his theorem on product of compact spaces.
		Using the notions of a compact topological space and the Tychonoff theorem, more elegant treatment of weak and weak* topologies, and the duality of Banach spaces was developed by L. Alaoglu, N. Bourbaki and S. Kakutani (1938--1940).
	arts-sciences.cua.edu /math/MIO/seminars/sem3_13.htm (315 words)

	MIMUW - Roman Pol (Site not responding. Last check: 2007-10-07)
		General topology and its relations to modern analysis and algebra, VI (Prague, 1986), pp.421-436, R and E Res.
		General topology and its relations to modern analysis and algebra, V (Prague, 1981), pp.548-560, Sigma Ser.
		The Proceedings of the 1980 Topology Conference (Univ. Alabama, Birmingham, Ala., 1980).
	www.mimuw.edu.pl /english/research/imat-publications/pol_roma.html (866 words)

	CVGMT: Relaxation of some nonlocal integral functionals in weak topology of Lebesgue spaces (Site not responding. Last check: 2007-10-07)
		in weak topology of a Lebesgue space L
		It is proven that, unlike the classical case without deviations, the relaxed functional in general cannot be obtained as convexification of the original one.
		Further slightly restricting this condition, we also obtain the nice representation of the relaxed functional in terms of convexification of some new integrand, but involving in general countably many new argument deviations.
	cvgmt.sns.it /papers/stezdo02 (172 words)

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