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Topic: Topological vector space

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	PlanetMath: topological vector space
		A topological vector space is necessarily a topological group: the definition ensures that the group operation (vector addition) is continuous, and the inverse operation is the same as multiplication by
		A finite-dimensional vector space inherits a natural topology.
		This is version 18 of topological vector space, born on 2002-02-03, modified 2007-07-04.
	www.planetmath.org /encyclopedia/TopologicalVectorSpace.html (152 words)

	PlanetMath: Fréchet space
		An F-space is a complete topological vector space whose topology is induced by a translation invariant metric.
		Recall that a topological vector space is a uniform space.
		A Fréchet space is a complete topological vector space (either real or complex) whose topology is induced by a countable family of semi-norms.
	planetmath.org /encyclopedia/FrechetSpace.html (436 words)

	Vector space Summary
		In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors.
		Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.
		Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors (quantities with a magnitude and a direction, usually depicted as arrows).
	www.bookrags.com /Vector_space (4366 words)

	Database Concepts
		Vector systems are capable of very high resolution (less than or equal to.001 inch) and graphical output is similar to hand-drawn maps.
		In a vector system, the locational symbol may be a one-dimensional point; a two-dimensional line, curve, boundary, or vector; or a three- dimensional area, region, or polygon.
		The starting point of the vector is referred to as the "from node" and the destination the "to node." The orientation of a given vector can be assigned in either direction, as long as this direction is recorded and stored in the database.
	www.colorado.edu /geography/gcraft/notes/datacon/datacon.html (3271 words)

	UC Davis Math: Glossary (Site not responding. Last check: )
		A high-dimensional plane or submanifold in a vector space or manifold with codimension 1.
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
		Given a topological space X, its loop space is the topological space of all continuous functions from a circle to X. Loop spaces are important examples of new topological spaces formed from old ones, as well as examples of infinite-dimensional spaces in mathematics.
	math.ucdavis.edu /profiles/glossary.html (9932 words)

	Topological Vector Space
		A topological vector space is a vector space with a topology, such that addition and scaling are continuous.
		A normed vector space is a topological vector space, deriving its topology from the metric.
		In a metric space, the translate of an open ball is an open ball, since the distance between two points does not change; but in a topological group, we have to use the properties of continuity to prove translation preserves open sets.
	www.mathreference.com /top-ban,tvs.html (1402 words)

	More on Vector Spaces
		A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
		Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right.
		The definition of a vector space makes perfectly good sense if one replaces the field of scalars F by a general ring R. The resulting structure is called a modules over R. In other words, a vector space is nothing but a module over a field.
	www.artilifes.com /vector-spaces.htm (1272 words)

	Banach space (Site not responding. Last check: )
		The space, together with this norm, is a Banach space; it is denoted by l p.
		A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology.
		Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces.
	encyclopedia.vestigatio.com /Banach_space (1317 words)

	Springer Online Reference Works
		Specific function spaces have been studied in detail, since the properties of these spaces usually determine the character of the solution to a problem when it is obtained by the methods of functional analysis.
		The so-called imbedding theorems for the Sobolev spaces
		However, for the majority of Banach spaces, and especially for topological vector spaces, the functionals are elements of a new kind which cannot be expressed simply in terms of classical analysis.
	eom.springer.de /F/f042020.htm (2924 words)

	Citations: Topological Vector Spaces - Schaefer (ResearchIndex)
		This is an analog of the notion of the orthogonal complements in Hilbert spaces.
		....L 1 (Q) closure of the vector space generated by G 0 is contained in f Delta S T : 2 L 1 loc (S; Q) such that Delta S is a Q martingaleg: According to Proposition 1.1 in [13] this result is valid without the assumption of a complete filtration.
		H.H. Schaefer, Topological Vector Spaces, Macmillan, New-York, 1966.
	citeseer.ist.psu.edu /context/109645/0 (2346 words)

	Reference.com/Encyclopedia/Local boundedness
		Local boundedness may also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space.
		A topological vector space is said to be locally bounded if X admits a bounded open neighborhood of 0.
		Let X be a topological space, Y a topological vector space, and f : X → Y a function.
	www.reference.com /browse/wiki/Local_boundedness (655 words)

	[No title]
		Representing a topological space by a finite polyhedron is not the only way to "discretise" the topological information in such a way that it can be dealt it by a computer.
		In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces.
		One method for studying topological spaces is to assign algebraic objects, such as groups or vector spaces, to a topological space.
	www.lycos.com /info/topological-space.html (410 words)

	Math through DSP: Basic linear algebra
		of a vector space is a subset of
		We conclude that a vector space can be isomorphic with its proper subspace and that given a linearly independent set of vectors with the cardinality equal to the dimension of the surrounding space, the set still might not be a basis.
		If we have two vector spaces over the same field, the set of all mappings between them can be made into a vector space by defining addition of mappings as the pointwise addition of their image vectors and multiplication by a scalar as pointwise multiplication of the result vectors by the scalar.
	www.helsinki.fi /~ssyreeni/dsound/dspmath-c-04 (2838 words)

	Hilbert space Summary
		In mathematics, a Hilbert space is a generalization of Euclidean space that is not restricted to finite dimensions.
		Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.
		Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces.
	www.bookrags.com /Hilbert_space (2249 words)

	Algebraic Topology: Homotopy
		Given two spaces X,Y, and a map f from X to Y, let [f] denote the homotopy class of f, that is, the set of all maps from X to Y homotopic to f.
		A topological space X is called simply connected when it is path-connected and its fundamental group P(X) is trivial.
		Let X be a topological space that is the union of two path-connected subspaces A and B, where the intersection of A and B is nonempty and path-connected.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	An Introduction to Banach Space Theory
		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.
		The first section includes some topological preliminaries, but is devoted primarily to a fairly extensive development of the theory of nets, including characterizations of topological properties in terms of the accumulation and convergence of certain nets.
		The section includes a short discussion of topological groups, primarily to be able to obtain a characterization of relative compactness in topological groups in terms of the accumulation of nets that does not always hold in arbitrary topological spaces.
	www.math.lsa.umich.edu /~meggin/ibst.html (2875 words)

	Definition of a Simplex
		A simplex exists in the context of a topological vector space.
		This is rather abstract, but a simplex actually lives in a finite dimensional subset of our topological space, and such a subset is equivalent to euclidean space.
		So you may picture a simplex as a structure in real space, which may in turn be part of a larger topological vector space.
	www.mathreference.com /top-sx,def.html (1215 words)

	Graduate Study in Geometry and Topology
		In a homogeneous space there is a distinguished group of differentiable mappings of the space into itself which acts transitively on points.
		An abstract space of such mappings is the prototype of a Lie group, so Math 522 and 507 are basic to their study.
		General topology has been an active research area for many years, and is broadly the study of topological spaces and their associated continuous functions.
	www.math.uiuc.edu /GraduateProgram/researchmath/gradgeomtop.html (1213 words)

	Orðasafn: T
		tangent space snertirúm, snertlarúm, = tangent vector space.
		topological homeomorhism grannmótun, = bicontinuous mapping, = homeomorphic mapping, = homeomorphism, = topological isomorphism, = topological mapping.
		topological linear space línulegt grannrúm, = linear topological space, = topological vector space.
	www.hi.is /~mmh/ord/safn/safnT.html (1134 words)

	PlanetMath: normed vector space
		It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.
		Cross-references: argument, convex function, inner product space, continuous map, induced, topology, locally convex topological vector space, metric, metric space, subspace, properties, triangle inequality, positive, function, vector space, field
		This is version 10 of normed vector space, born on 2002-01-24, modified 2006-12-08.
	planetmath.org /encyclopedia/NormedVectorSpace.html (249 words)

	Topological Space
		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.
	jcbmac.chem.brown.edu /baird/QuantumPDF/space.html (566 words)

	prime-space - a real topological vector space, used to model musical tunings
		Prime-space refers specifically to the objectification of the pitch-continuum in multi-dimensional space as a series of axes, each of which represents simultaneously one unique prime-factor and one unique dimension of space, and along each of which are equally-spaced points representing the exponents of those prime-factors.
		and not an infinite-dimensional space, since then this definition is precise--the vector space "inherits" a topology (product topology) from the real numbers as a topolgical field.
		Since a lattice can be defined in a topological vector space (as a discrete subgroup), this definition gives a precise, but rather abstract, sense in which the lattice of monzos for any given prime limit is uniquely defined.
	tonalsoft.com /enc/p/prime-space.aspx (587 words)

	HAF: Preface
		Moreover, it may confuse beginners by entangling concepts that are not inherently related: The basic ideas of Hausdorff spaces are independent from the other basic ideas of uniform spaces, topological spaces, and locally convex spaces; neither set of ideas actually requires the other.
		For instance, convexity is commonly introduced in functional analysis courses in the setting of Banach spaces or topological vector spaces, but I have found it expedient to introduce convexity as a purely algebraic notion, and then add topological considerations much later in the book.
		Most topological vector spaces used in applications are locally convex, but HAF first studies topological vector spaces without the additional assumption of local convexity.
	www.math.vanderbilt.edu /~schectex/ccc/excerpts/preface.html (4201 words)

	Annotated Bibliography on the Range of Vector Measures (Site not responding. Last check: )
		Ferakova and Nãther (1988) proved that the infinite-dimensional Lyapunov theorem for vector measures defined on a sigma-algebra is valid also in the case that the vector measures are defined only on a sigma-ring.
		Maritz (1980-81) showed that the bilinear integral of a set-valued function with values in an arbitrary Banach space is convex, provided the integral is a subset of a finite dimensional space and the measure is atomless.
		If mu is an atomless finite-dimensional vector measure (not necessary finite), then the range of mu is convex, the closure of the range does not contain a line and each compact extreme face of the closure of the range is contained in the range.
	www.math.gatech.edu /~hill/publications/annotated.html (5995 words)

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