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Topic: Separable space

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	NationMaster - Encyclopedia: Separable space
		Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.
		The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem).
		A metric space is separable iff it is second-countable and iff it is Lindelöf.
	www.nationmaster.com /encyclopedia/Separable-space (2087 words)

	PlanetMath: separable space
		A topological space is said to be separable if it has a countable dense subset.
		This is version 5 of separable space, born on 2002-01-03, modified 2007-07-31.
		Object id is 1193, canonical name is Separable.
	planetmath.org /encyclopedia/Separable.html (57 words)

	Separable space (Site not responding. Last check: )
		Separable spaces are therefore topological spaces with a certain limit on their size: an uncountable discrete space isn't separable.
		Most of the spaces initially encountered are indeed separable: for example the real numbers with their standard metric have the rational numbers as a countable dense subset.
		Since the space of continuous functions on the interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximation theorem), and their coefficients can be approximated by rationals, that space is also separable.
	www.centipedia.com /articles/Separable_(topology) (384 words)

	Separable space: Definition and Links by Encyclopedian.com
		In topology and related branches of mathematics, separable spaces are topological spaces with a certain limit on their size.
		To be specific, a topological space is separable if and only if it has a subset that is both countable and dense.
		Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.
	www.encyclopedian.com /se/Separability.html (400 words)

	Proceedings of the American Mathematical Society
		Arhangel'skii A.V., Structure and classification of topological spaces and cardinal invariants, Russian Math.
		Asanov M., Cardinal invariants of spaces of continuous functions, Modern Topology and Set Theory, Izhevskij Universitet, Izhevsk, 1979, pp.
		Hajnal A. and I. Juhasz, A separable normal topological group need not be Lindelöf, General Topology and Appl.
	e-math.ams.org /proc/1996-124-11/S0002-9939-96-03605-2/home.html (389 words)

	Space-time separable RF (Site not responding. Last check: )
		The number that appears at the bottom left corner of the animation is the time (correlation delay) in milliseconds at which the spatial (X-Y) map is displayed.
		Horizontal axis is space (6 degs), vertical axis is time (0-300 msec).
		Cells that have space-time separable RFs are generally not direction selective, i.e., they respond equally well to both forward and reverse movements of stimuli.
	www.bpe.es.osaka-u.ac.jp /ohzawa-lab/teaching/RF/XTseparable.html (292 words)

	To Clear a Space by Hugh Shearman
		Space is universal and infinite, and if space is present all is present.
		True space is at one with that which anywhere contains it and so contributes subtly to shaping the vessel which contains it; but when a container is devised more as a trap than as an open shrine for space, and the space itself is conceptual rather than experienced, then the meditation leads into unreality.
		This space divides man from man, and in it there is all the becoming, the battle of life, the agony and fear.
	www.tphta.ws /HSH_TCAS.HTM (1602 words)

	20th WCP: The Construction of Juridical Space: Kant's Analogy of Relation in the Metaphysics of Morals
		Hence we may invoke the third analogy construction of space as an illustration of the compossibility of movements or actions under law, established by mutual limitation or coercion.
		This relation among bodies is not simply a formulation of Newton's law of action and reaction, but refers to interactions in space, in which bodies resist penetration by others up to their limits, and are equally repelled from other bodies.
		The unity and continuity of space are constituted by this mutuality, or the congruity of limits.
	www.bu.edu /wcp/Papers/TEth/TEthMogg.htm (3149 words)

	Article 3
		Merleau-Ponty by opposing Kant's concept of space as a pure space, "in the mind", argues that spatiality is integrally associated with "Lived-Body" as a joint contributor and embedded in the existence.
		The space is divided into 18 seemingly identical cubes by use of stretched scrim, acting as "walls" and attached to a hidden framework.
		Space is like the Cezanne's table cloth in the still life paintings a "condition" for the possibility of things, (apples), having position in connection to other things.
	www.ehusserl.com /article3.htm (8404 words)

	Kit Chan's Annotated Bibliograhy
		On a separable infinite dimensional complex Hilbert space, we show that the set of hypercyclic operators is dense in the strong operator topology, and moreover the linear span of hypercyclic operators is dense in the operator norm topology.
		We show that the translation operator on a Hilbert space of entire functions of slow growth is hypercyclic, which means that there is a function in the space whose translates are dense.
		For the Dirichlet space of a finitely connected region, we study the structure of its subspaces that are invariant under the algebra of multiplication operators.
	personal.bgsu.edu /~kchan/docs/biblio.html (1091 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.
		These are function spaces associated to measure spaces (X, M, μ), where M is a σ-algebra of subsets of X and μ is a countably additive measure on M.
	www.mathdaily.com /lessons/Hilbert_space (1716 words)

	S.O.S. Mathematics CyberBoard :: View topic - Separable space?
		This is a special case of a more general theorem, which I guess is unknown to most students and would probably not have sufficed as an answer.
		Now, one could (and perhaps should) view IR^I as the space of functions defined on the closed unit interval, taking values in IR.
		I hope you agree there is such a basis (a simple consequence of Lindelöf's theorem, since separable metric spaces are also second countable, but in an explicit case like this you can think of open intervals having rational end-points, say).
	www.sosmath.com /CBB/viewtopic.php?t=26923 (533 words)

	Math Forum Discussions
		A topological space is said to be "separable" if it has a countable dense
		In particular, the space R^n is separable for each n.
		example, the continuous image of a separable space is separable.
	www.mathforum.org /kb/message.jspa?messageID=4769203&tstart=0 (484 words)

	Banach space preliminaries
		The purpose of this section is to review some well-known concepts and results from separable Banach space theory.
		In particular, the weak-* topology is not in general metrizable, even when X is separable.
		is a closed subset of a complete metric space, written as a countable union of closed sets.
	www.math.psu.edu /simpson/papers/convex-l/node2.html (999 words)

	Space Politics » To Infinity and Beyond (Site not responding. Last check: )
		And John Stennis, the patronage hero of Stennis Space Center, was a Democrat.
		It’s a reasonable policy to replace an aging vehicle, the space shuttle, with a safer one.
		It’s a reasonable policy to build the shuttle replacement with as much existing technology as possible, to reduce the risks of untried new technologies (which was the mistake that earlier shuttle replacement programs had).
	www.spacepolitics.com /archives/000540.html (3098 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)

	A Universal Separable Metric Space Based on the Triangular Sierpinski Curve (Site not responding. Last check: )
		A year later he constructed his rectangular curve and proved that is universal for planar one-dimensional spaces.
		The rectangular curve is a particular case of today called Menger spaces.
		The subset of this product which consists of points having at least one irrational coordinate is a universal space for the class of all separable metrizable spaces of dimension less than or equal n.
	www.pmf.ukim.edu.mk /mathematics/ivansic.htm (186 words)

	Quantum Theory: von Neumann vs. Dirac (Stanford Encyclopedia of Philosophy)
		The trace of the identity is infinite when the separable Hilbert space is infinite-dimensional, which means that it is not possible to define a correctly normalized a priori probability for the outcome of an experiment (i.e., a measurement of an observable).
		In the mathematical theory of topological vector spaces, the space Φ is characterized in technical terms as a nuclear Fréchet space.
		In distribution theory, the space Φ is characterized as a test-function space, where a test-function is thought of as a very well-behaved function (being continuous, n-times differentiable, having a bounded domain or at least dropping off exponentially beyond some finite range, etc).
	plato.stanford.edu /entries/qt-nvd (10281 words)

	FuncAna
		Dual space of normed vector space is a Banach space.
		Local weak compactness of the dual spaces of normed vector spaces: closed bounded sets in the dual space are weakly compact.
		Orthogonal sum decomposition of the space in terms of the null spaces and ranges of an operator and its adjoint.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	The Hierarchy of Borel Universal Sets
		Each section is further divided in two, with the first part dealing with general spaces, and the second the special case of compact spaces.
		In particular, a space is separable metrisable if and only if parametrised by a separable metrisable space.
		All spaces are regular Hausdorff topological spaces unless stated otherwise.
	pear.math.pitt.edu /mathzilla/Examples/DynHieruni.xml (1264 words)

	The Second Rank Contravariant 4-Space Tensor
		The second rank 3+1 space contravariant tensor is derived below.
		Does that mean that the expansion factor of 4-space is different than the expansion factor of 3+1 space?
		This tensor seems to have the same meaning as the 3+1 space tensor shown above, and the 4-space version is easier to work with.
	www.s-4.com /tensor/tensor (225 words)

	Glossary of Topology Terms
		a topological space is compact if every collection of open sets that covers the space has a finite subset that also covers the space.
		Two points lie in the same path component of a space X iff there is a path in X from one point to the other.
		a separable space is one that has a countable dense subset, that is a countable subset whose closure is the whole space.
	cage.rug.ac.be /~hvernaev/TopoGloss.html (2536 words)

	Closure of an open ball and separable space?
		a discrete metric space is just the example you're in need of.
		2)A separable space is a metric space with a
		it is true that in a discrete metric space one can find open blass of form: {all points closer than r to p} whose closure is not the closed ball {all points at distance at most r from p}.
	www.physicsforums.com /showthread.php?p=382517 (631 words)

	Separable Hilbert Space in Loop Quantum Gravity
		In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable.
		The space of these maps form a group under composition, because the composition of two homeomorphisms that are smooth except at a finite number of singular points is clearly an homeomorphisms which is smooth except at a finite number of singular points.
		In this paper by Fairbairn/Rovelli the quantum state of all space is denoted by a knot and a quantum number.
	www.physicsforums.com /showthread.php?p=166533 (3281 words)

	Springer Online Reference Works
		Quotient spaces and complemented subspaces of a Grothendieck space are also Grothendieck spaces.
		Reflexive Banach spaces are obvious examples of Grothendieck spaces (cf.
		Every separable quotient space of a Grothendieck space is necessarily reflexive.
	eom.springer.de /G/g110250.htm (351 words)

	GeomFest 2006 / Department of Mathematics at the University of Pennsylvania (Site not responding. Last check: )
		In recent joint work with Bruce Kleiner, we extended the differentiation and nonembedding results to infinite dimensional Banach space targets, which are separable and dual to some Banach space.
		These invariants are constructed from the solution spaces of a countable number of nonlinear Cauchy-Riemann type problems.
		With these we are able to compute the topological dimension of (locally compact subsets of) the cone, and hence establish Brock-Farb's geometric rank conjecture, which states that the maximal dimension of a quasi-isometrically embedded Euclidean space in the group is equal to the maximal rank of an abelian subgroup.
	www.math.upenn.edu /geomfest/abstracts.html (913 words)

	Workshop on the Urysohn Space
		The Urysohn space U is the universal complete metric separable space.
		An interesting approach to the Urysohn space was proposed by A.
		We acknowledge the company Atlas Conferences as a workshop sponsor.
	www.math.bgu.ac.il /~arkady/Workshop_2006/main_page.html (333 words)

	AGI Research - Leading Edge COTS Software Solutions for Aerospace, Intelligence, and Space Initiatives
		Three-dimensional visualization of position and attitude information for satellites and other vehicles is highly beneficial to learning the concepts and applications of astrodynamics and attitude determination and control.
		The TLE sets used are maintained and updated in the space catalog, which has been maintained by various military organizations at the Cheyenne Mountain Complex in Colorado Springs, CO, since 1965.
		Though correlated, atmospheric density modeling errors and ballistic coefficient modeling errors are shown to be observable and separable.
	www.agi.com /research/research.html (1953 words)

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