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History of the Fundamental Theorem of Calculus Cauchy defined the integral of any continuous function on the interval [a,b] to be the limit of the sums of areas of thin rectangles. Cauchy proved the Mean Value Theorem for Integrals and used it to prove the Fundamental Theorem of Calculus for continuous functions, giving the form of the proof used today's calculus texts. Cauchy the first to define fully the ideas of convergence and absolute convergence of infinite series, including the development of the ratio and root tests for convergence of series. www.saintjoe.edu /~karend/m441/Cauchy.html   (1146 words)

Cauchy spaces and proximity spaces, by Gerhard Preuss   (Site not responding. Last check: 2007-09-15) Cauchy spaces and proximity spaces, by Gerhard Preuss Cauchy spaces are obtained by axiomatizing the concept of Cauchy filter which is fundamental for studying completeness. They are the suitable framework for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, namely simple convergence, continuous convergence and uniform convergence, which are very important for investigations in topology and analysis. at.yorku.ca /z/a/a/b/13.htm   (335 words)

Complete space - Wikipedia, the free encyclopedia In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. That is, the interior of a union of countably many nowhere dense subsets of the space is empty. The original space is embedded in this space via the identification of an element x of M with the equivalence class of sequences converging to x (i.e. en.wikipedia.org /wiki/Complete_space   (1200 words)

Banach space Summary This means that a Banach space is a vector space V over the real or complex numbers with a norm ‖·‖ such that every Cauchy sequence (with respect to the metric d(x, y) = ‖x − y‖) in V has a limit in V. The space V* (which may be called the algebraic dual space to distinguish it from V') also induces a weak topology which is finer than that induced by the continuous dual since V′⊆V*. 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. www.bookrags.com /Banach_space   (2223 words)

Real number : Real   (Site not responding. Last check: 2007-09-15) More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion). Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. www.termsdefined.net /re/real.html   (2470 words)

Cauchy space - Wikipedia, the free encyclopedia In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. Any Cauchy space is also a convergence space, where a filter F converges to x if F∩U(x) is Cauchy. en.wikipedia.org /wiki/Cauchy_space   (274 words)

Real numbers 2 Though Cauchy understood that a real number could be obtained as the limit of rationals, he did not develop his insight into a definition of real numbers or a detailed description of the properties of real numbers. Certainly this is not considered by Cauchy to be a definition of a real number, rather it is simply a statement of what he considers an "obvious" property. Cauchy himself does not seem to have understood the significance of his own "Cauchy sequence" criterion for defining the real numbers. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Real_numbers_2.html   (2626 words)

Cauchy sequence - Article from FactBug.org - the fast Wikipedia mirror site In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. Cauchy sequences require the notion of distance so they can only be defined in a metric space. They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence. www.factbug.org /cgi-bin/a.cgi?a=6085   (598 words)

PlanetMath: Cauchy-Schwarz inequality Such spaces can be given also a norm by defining That is, the modulus (since it might as well be a complex number) of the inner product for two given vectors is less or equal than the product of their norms. Kantorovich inequality, Bunyakovsky inequality, Schwarz inequality, Cauchy inequality, CBS inequality planetmath.org /encyclopedia/CauchySchwarzInequality.html   (274 words)

Springer Online Reference Works   (Site not responding. Last check: 2007-09-15) In a Hilbert space exponential stability is equivalent to For almost-periodic solutions the specifics of an infinite-dimensional space are already encountered in a generalization of the well-known Bohl–Bohr theorem on the almost-periodicity of a bounded integral of an almost-periodic function, that is, on the almost-periodicity of a solution of the simplest differential equation ) is homeomorphic to the neighbourhood of the space eom.springer.de /q/q076250.htm   (1687 words)

Hilbert Space Explorer Home Page   (Site not responding. Last check: 2007-09-15) Hilbert space [external] underlies the foundation of quantum mechanics, and there is a strong physical and philosophical motivation to understand its properties. The next 11 axioms are the axioms for any vector space with an unspecified dimension; they are the same as those you would find in any linear algebra book, except for the notation and possibly their precise form. The set of closed subspaces of Hilbert space obey the laws of a simple equational algebra called "orthomodular lattice algebra." This algebra is sometimes called "quantum logic," and it can be used as the basis for a propositional calculus that resembles but is somewhat weaker than standard (classical) propositional calculus. us.metamath.org /mpegif/mmhil.html   (2114 words)

Banach Space | World of Mathematics A Banach space is a vector space over the field of real numbers, or over the field of complex numbers, together with a norm. It consists of the vector space X over the field of real numbers or complex numbers, the norm function ∥ ∥, and the metric topology in X which is induced by the norm. We are now in position to give the definition of a Banach space: a Banach space is a normed linear space over the field of real numbers or over the field of complex number, such the metric determined by the norm is complete. www.bookrags.com /research/banach-space-wom   (958 words)

Personal space - Uncyclopedia, the content-free encyclopedia An important concept in Functional Analysis, Personal spaces were first developed by the famed mathemagician Big Norman, though it is now credited to Cauchy who has proved using Proof by intimidation that it was actually all done by him wearing a different hat. As any slack-jawed yokel knows, Normed Spaces are spaces of thingies equipped with a John Norman novel, or 'Norm'. A proof is yet to arrive from Cauchy, but he has assured the international community of mathemagicians that it's really easy, and "you would finish it in a snap if you were as good as I am." uncyclopedia.org /wiki/Personal_space   (574 words)

Cauchy sequence - The limit space of a Cauchy sequence of globally hyperbolic spacetimes Cauchy sequence - The limit space of a Cauchy sequence of globally hyperbolic spacetimes The sequence is Cauchy if and only if it converges to some limit a. The limit space of a Cauchy sequence of globally hyperbolic spacetimes northeastern.surferlight.com /?q=northeastern-cauchy-sequence   (145 words)

University of Pittsburgh: Department of Mathematics   (Site not responding. Last check: 2007-09-15) A separable space X is countable dense homogeneous (CDH) if, for every pair A,B of countable dense subsets of X, there is a homeomorphism of X onto itself that maps A onto B. Thus, all countable dense subsets of X are `positioned' in X in the same way. The intent of the project was to study the topological properties of the graphs of discontinuous real-valued additive functions (Cauchy spaces), especially discontinuous additive functions with connected graphs (Jones spaces), and especially the homogeneity properties of such functions. On the other hand the simplest Cauchy space (a graph of an additive function with only rational values) is not CDH. www.math.pitt.edu /summer2001-autoh.html   (580 words)

Cauchy Schwarz Inequality, Triangular Inequality Finite products of arbitrary metric spaces produce a valid metric, and when this general result is applied to coordinate axes you get n dimensional space, with its euclidean metric. The Cauchy Schwarz inequality is equivalent to the triangular inequality, at least in n space. The distance functions in the two spaces give exactly the same answer; both give the square root of the sum of the squares of the components. www.mathreference.com /top-ms,csi.html   (1704 words)

Volume 15 Abstracts The paper deals - under the viewpoint of topology - with discrete Cauchy spaces, which are spaces where a discrete Cauchy structure (t,C) (with t being a discrete convergence and C being a discrete pre-Cauchy structure) is defined. The construction of a completion of a discrete Cauchy space differs (in some sense essentially) from the construction of a completion of a usual sequential Cauchy space and is even more simple. A further subject of the paper are metric discrete Cauchy spaces of mappings between metric discrete Cauchy spaces, where simple characterizations of the corresponding discrete convergence and discretee pre-Cauchy structure of such a discrete Cauchy space as well as a necessary and sufficient condition for its completeness are given. www.heldermann.de /ZAA/zaaabs15.htm   (2320 words)

Vector Space Concepts is the space of continuous complex functions on the unit circle in the Banach Space is a complete normed linear space, that is, a normed linear space in which every Cauchy sequence A Hilbert space is a Banach space with a symmetric bilinear www-ccrma.stanford.edu /~jos/gradient/Vector_Space_Concepts.html   (225 words)

[No title]   (Site not responding. Last check: 2007-09-15) The target space $M\cfg$ of configurations has to be chosen such that the image of $\check\Phi\sol$ still is the set of all classical solutions in $M$. Turning to the functional spaces needed, a reasonable choice for the Cauchy data is the test function space $\cEc\Cau:=\cD(\rdm)$; for the configurations we take the space $\cEc$ of all those $f\in C^\infty(\rdmm)$ the support of which on every time slice is compact and grows only with light velocity (cf. If $E$, $F$ are spaces of generalized functions on $\rdm$ which contain the test functions as dense subspace then the Schwartz kernel theorem tells us that the multilinear forms $u_{(kl)}$ are given by their integral kernels, which are generalized functions. www.ma.utexas.edu /mp_arc/papers/96-340   (11330 words)

PlanetMath: Cauchy sequence is a Cauchy sequence if and only if for every neighborhood Cross-references: metric, induced, topology, equivalent, neighborhood, topological vector space, natural number, real number, metric space, sequence This is version 5 of Cauchy sequence, born on 2001-10-27, modified 2005-11-30. planetmath.org /encyclopedia/CauchySequence.html   (93 words)

Cauchy Sequences   (Site not responding. Last check: 2007-09-15) A sequence s in a metric space is Cauchy (biography) if every ε implies an n such that all points beyond s Set ε to 1 to show that a Cauchy sequence, or the tail of a Cauchy function, is bounded. After all, metric spaces are based on distance, which is measured using real numbers, so we'd better get a handle on the reals first. www.mathreference.com /top-ms,cauchy.html   (241 words)

Function spaces All of the examples from §2 are complete function spaces. is a subset of the space of distributions. Sobolev spaces are Banach spaces where the norm involves derivatives, or at least, something other than just function values. www.math.uiowa.edu /~dstewart/classes/22m176/dfs-notes/node2.html   (605 words)

Cauchy.html It is also the case that Cauchy sequences are not preserved under mapping by continuous functions. There is an important sub-class of continuous functions which do preserve Cauchy sequences and, in fact, are the continous functions on an important sub-Category of Complete Metric Spaces. To verify that complete metric spaces and uniformly continuous maps form a category we need to check the the composition of uniformly continuous maps is uniformly continuous. www.umsl.edu /~siegel/SetTheoryandTopology/Cauchy.html   (236 words)

Hilbert Space   (Site not responding. Last check: 2007-09-15) As the example above shows, the space of rational numbers, with the usual notion of distance, is not a complete metric space. I1, as a metric space with a "distance between functions f and g" defined by You should now compare these representations with those for a finite dimensional vector space, and convince yourself that these two sets are formally identical. www.chem.brown.edu /chem277/Tan_on_Hilbert_Space.html   (1211 words)

[No title]   (Site not responding. Last check: 2007-09-15) ] The conditions imposed on a surface in euclidean space which are to be satisfied by a solution to a partial differential equation. ] The Cauchy data for a hyperbolic partial differential equation consist of the value of the field and its time derivative on some spacelike surface. ] A net whose members are elements of a topological vector space and which satisfies the condition that for any neighborhood of the origin of the space there is an element a of the directed system that indexes the net such that if b and c are also members of this directed system and b www.accessscience.com /Dictionary/C/C12/DictC12.html   (1783 words)

MATH 302-11 Ordinary Differential Equations Winter 2002   (Site not responding. Last check: 2007-09-15) Definition and examples of metric spaces - R^n, C, set of real valued bounded functions on a set E. Prove $R^n$ is a metric space - using Cauchy-Schwartz inequality. Limit of a sequence in a metric space. Def of continuity of functions on metric spaces. www.math.udel.edu /~cakoni/Syllabus-M600.html   (159 words)

LiteMat Titel: On uniqueness of slutions to polyharmonic equations in the half-space with Cauchy data and imcomplete Cauchy data. Abstract: In this thesis we study uniqueness of solutions to a biharmonic equations in the upper half-space with incomplete Cauchy data. Moreover, this is not a Cauchy problem since we don't have any third order Cauchy data on $G$. www.mai.liu.se /LiteMat/2005/v33-05   (281 words)

Volume 24, Number 1, 1998 It is shown that the construct of supertopological spaces and continuous maps is topological. In particular it follows from Künzi's [8] proofs that each totally bounded locally quiet quasi-uniform space is uniform, and recently Déak [10] observed that even each totally bounded Cauchy quasi-uniformity is a uniformity. A new class of regular spaces, called CE-regular spaces, is introduced; the class of all OCE-regular spaces of J. Porter and C. Votaw [29] (and, hence, the class of all regular-closed spaces) is its proper subclass. www.math.bas.bg /~serdica/n1_98.html   (1107 words)

Mathematical Structure -- Inner Product Spaces   (Site not responding. Last check: 2007-09-15) In many vector spaces but not all vector spaces we can talk about geometric ideas like length and angles. Check that this definition of magnitude agrees with the definitions we gave for all vector spaces discussed in the module on magnitude. The inequality in this theorem is often called the triangle inequality because it is an immediate consequence of the original triangle inequality and because it says that the direct distance between two vertices x and z of a triangle is less than or equal to the distance by way of the third vertex, y. www.math.montana.edu /frankw/ccp/multiworld/building/dotproduct/refer.htm   (1194 words)

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