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Topic: Bounded subset

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	Compactness
		A compact subset of R with its usual metric is closed and bounded.
		Any closed bounded subset of R with its usual metric is compact.
		The closed bounded interval is compact and hence its image is compact and hence is also a closed bounded subset which is in fact an interval also, by connectedness.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L21.html (704 words)

	PlanetMath: totally bounded subset of a metric space is bounded
		be a totally bounded subset of a metric space.
		"totally bounded subset of a metric space is bounded" is owned by georgiosl.
		This is version 8 of totally bounded subset of a metric space is bounded, born on 2005-07-26, modified 2006-10-14.
	www.planetmath.org /encyclopedia/TotallyBoundedSubsetOfAMetricSpaceIsBounded.html (102 words)

	Kids.Net.Au - Encyclopedia > Bounded
		The term bounded appears in different parts of mathematics where a notion of "size" can be given.
		A set S in a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e.
		A set S in a topological vector space is bounded if it is contained in some multiple of every basic neighbourhood of zero.
	www.kids.net.au /encyclopedia-wiki/bo/Bounded (254 words)

	NationMaster - Encyclopedia: Spectrum of an operator
		In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, fails badly to be invertible.
		A bounded operator may be viewed as an element of a Banach algebra, with the definition of spectrum transferred verbatim from that context.
		The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
	www.nationmaster.com /encyclopedia/Spectrum-of-an-operator (1673 words)

	NationMaster - Encyclopedia: Nuclear operator
		In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite.
		In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a...
		In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2.
	www.nationmaster.com /encyclopedia/Nuclear-operator (773 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In mathematics, bounded variation refers a to real-valued functions whose total variation is bounded i.e.
		Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure.
		Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=bounded_variation (989 words)

	Totally bounded space - Definition, explanation
		A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E.
		A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and only if, given any neighbourhood E of the identity (zero) element of X, there exists a finite cover of S by subsets of X each of which is a translate of a subset of E.
		A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
	www.calsky.com /lexikon/en/txt/t/to/totally_bounded_space.php (1208 words)

	SCS
		One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen.
		A subset of a totally bounded set is again totally bounded iff it is located.
		Moreover, a closed subset of a complete metric space is Bishop compact --- that is, totally bounded and complete --- iff its localic completion is compact overt.
	calendar.cs.cmu.edu /scsEvents/demo/3417.html (236 words)

	Bounded
		The term bounded appears in different parts of mathematics where a notion of "size" can be given.
		A set S in a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e.
		A set S in a topological vector space is bounded if it is contained in some multiple of every basic neighbourhood of zero.
	www.ebroadcast.com.au /lookup/encyclopedia/bo/Bounded.html (0 words)

	Functionally bounded subsets of topological spaces, by Manuel Sanchis Lopez (Site not responding. Last check: )
		Functionally bounded subsets and some related concepts arose, in a natural way, during the study of the topological properties induced by real-valued continuous functions.
		The denomination functionally bounded has been used in the theory of Hausdorff topological groups (J.Trigos-Arrieta, 1991) to distinguish between this concept and the concept of the bounded subset in the sense of precompact subset.
		These subsets play an important role in the field of Hausdorff topological groups: every functionally bounded subset of a Hausdorff topological group is strongly bounded.
	at.yorku.ca /z/a/a/b/02.htm (425 words)

	T-F , Quantifiers
		Every bounded subset of Z has a least element.
		The sum of two bounded sequences is bounded.
		The intersection of any sequence of bounded nested intervals is nonempty.
	www.fiu.edu /~hudsons/m32/m32exams/tfWade1.htm (126 words)

	Totally bounded space - Wikipedia, the free encyclopedia
		A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E.
		A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and only if, given any neighbourhood E of the identity (zero) element of X, there exists a finite cover of S by subsets of X each of which is a translate of a subset of E.
		A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
	en.wikipedia.org /wiki/Totally_bounded_space (0 words)

	Definitions
		The bounding polygon of a set of points in the plane is the polygon of smallest area that contains all of the given points.
		The convex hull of a bounded subset of a 2D plane is the convex set of smallest area that contains the original set.
		A bounded subset of a 2D plane is convex if, for any two points in the set, all points on the line segment between the two points are in the set.
	ngwww.ucar.edu /ngdoc/ng/ngmath/definitions.html (1295 words)

	Content
		The existence of representation by Dedekind-cuts of a dense subset is equivalent to separability of a metric space.
		A subset of R is compact if and only if it is closed and bounded.
		Supremum and infimum may not be achievable for a bounded function f defined on [a,b]; that is, there may not exist a point x_0 in [a,b] such that f(x_0)=sup f(x) or f(x_0)=inf f(x).
	www.cs.cmu.edu /~dpwu/books/math/analysis/RealAnalysis.html (3085 words)

	Wikinfo \| Metric space (Site not responding. Last check: )
		A subset of M which is a union of (finitely or infinitely many) open balls is called an open set.
		A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely).
		An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem).
	www.wikinfo.org /wiki.php?title=Metric_space (0 words)

	Overview of Field Computation
		A field is treated mathematically as a continuous function over a bounded set representing the spatial extent of the field.
		Typically, the value of the function is restricted to some bounded subset of the real numbers, but complex- and vector-valued fields are also useful.
		I have already mentioned that fields are continuous functions over a bounded domain that take their values in a bounded subset of a linear space.
	www.cs.utk.edu /~mclennan/anon-ftp/FCNAI-summary-TR/node2.html (609 words)

	Reference.com/Encyclopedia/Local boundedness
		A family of functions is locally bounded, if for any point in their domain all the functions are bounded around that point and by the same number.
		In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant.
		This family is then more than locally bounded, it is actually uniformly bounded.
	www.reference.com /browse/wiki/Local_boundedness (655 words)

	Bounded set - Wikipedia, the free encyclopedia
		In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.
		A subset S of a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e.
		M is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself.
	en.wikipedia.org /wiki/Bounded_set (0 words)

	[No title] (Site not responding. Last check: )
		The "compact iff closed and bounded" property is confined to finite dimensional spaces.
		Otoh if V is a topological vector space there is a standard notion of "bounded subset"; if V is a metrizable topological vector space then a bounded subset is not at all the same thing as a subset which is bounded as a subset of the corresponding metric space.
		This is unfortunate because it can cause confusion - people sometimes think that a bounded subset of V is a subset of finite diameter in "the" metric.
	www.math.niu.edu /~rusin/known-math/00_incoming/heine (526 words)

	Body
		In a generalisation of area under a graph of a function it is normally assumed that the function under consideration be bounded.
		For bounded function the range of the function is bounded and hence any subset of the range is also bounded.
		Because subset not bounded above does not have supremum and subset not bounded below does not have infimum, for unbounded functions lower and upper Riemann sums cannot be defined.
	www.math.nus.edu.sg /~matngtb/Calculus/Riemann/Riemann.htm (1526 words)

	Bounded Complex Domains
		What the Shilov boundary is good for is that it is the minimal subset of the topological boundary over which you can integrate an analytic function to reproduce its interior values by the Poisson kernel.
		The compact manifold that represents 4-dim internal symmetry space is CP2, the Shilov boundary of the bounded complex homogeneous domain that corresponds to SU(3) / (SU(2)xU(1)).
		The Shilov boundary of an irreducible bounded symmetric domain is a flag manifold of the 1st kind or of the 2nd kind, according as the domain is of tube type or not.
	valdostamuseum.org /hamsmith/cdomain.html (8754 words)

	Compactness
		A compact subset of R with its usual metric is closed and bounded.
		Any closed bounded subset of R with its usual metric is compact.
		The closed bounded interval is compact and hence its image is compact and hence is also a closed bounded subset which is in fact an interval also, by connectedness.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L21.html (704 words)

	[No title]
		Thanks for enlightening me. Keep in mind that "bounded" only makes sense in a metric space, and that "closed" implies X is a subset of something else (evey topological space is closed in itself).
		Thus for subsets of a _complete_ metric space, X is compact iff it is closed and _totally_ bounded.
		Totally bounded (or "pre-compact") means that for every positive epsilon there exists a finite epsilon-dense subset (or equiv., there exists a finite cover by sets of diameter < epsilon).
	www.math.niu.edu /~rusin/known-math/95/compact.nss (1036 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		This terminology may be traced to the well-known fact that the open-set lattice for the prime spectrum of a bounded, distributive lattice is a coherent frame (see Johnstone [13]).
		P is order-isomorphic to the prime spectrum of a bounded, distributive lattice in which every element is the join of a finite set of join-prime elements.
		P is order-isomorphic to the prime spectrum of a bounded, relatively normal lattice in which every element is the join of a finite set of join-prime elements.
	www.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	FuncAna
		Local weak compactness of the dual spaces of normed vector spaces: closed bounded sets in the dual space are weakly compact.
		Remark: the kernel of a bounded operator is always closed while the range may be not closed.
		A bounded operator is an orthogonal projector if and only if it is idempotent and self-adjoint.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (0 words)

	The Group of Units
		is closed, bounded, symmetric, etc., and has volume at least
		is closed, bounded, convex, symmetric with respect to the origin, and of dimension
		The amazing thing about (12.1.4) is that the bound
	modular.fas.harvard.edu /papers/ant/html/node41.html (453 words)

	Continuity and All That
		If B is compact, then as E is a closed subset of a compact set, is also compact.
		As E is a closed subset of B, then E is compact.
		E is a closed subset but not compact, then E cannot be compact.
	cepa.newschool.edu /het/essays/math/contin.htm (1663 words)

	Chow, Theresa Kee Yu (1969-04-07) The Egoroff property and its relation to the order topology in the theory of Riesz ...
		The pseudo order closure S' of a subset S is the set of all [...] such that there exists a sequence in S which is order convergent to f.
		The pseudo ru-closure S'[subscript ru] of a subset S is the set of all [...] such that there exists a sequence in S which is ru-convergent to f.
		If L is Archimedean, then [...] for every convex subset S implies that [...] for every subset S. A characterization of those Archimedean Riesz spaces L with the property that [...] for every subset S of L is obtained.
	etd.caltech.edu /etd/available/etd-10072002-143502 (407 words)

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