Definition in Terms of Order Alone in the Linear
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Boundedness in Linear Topological Spaces
BOUNDEDNESS IN LINEAR TOPOLOGICAL SPACES BY S. SIMONS Introduction. Throughout this paper we use the symbol X for a (real or complex) linear space, and the symbol F to represent the basic field in question. We write R+ for the set of positive (i.e., ^ 0) real numbers. We use the term linear topological space with its usual meaning (not necessarily Tx), but we exclude the case where the space has the indiscrete topology (see [1, 3.3, pp. 123-127]). A linear topological space is said to be a locally bounded space if there is a bounded neighbourhood of 0—which comes to the same thing as saying that there is a neighbourhood U of 0 such that the sets {(1/n) U} (n = 1,2,—) form a base at 0. In §1 we give a necessary and sufficient condition, in terms of invariant pseudo- metrics, for a linear topological space to be locally bounded. In §2 we discuss the relationship of our results with other results known on the subject. In §3 we introduce two ways of classifying the locally bounded spaces into types in such a way that each type contains exactly one of the F spaces (0 < p ^ 1), and show that these two methods of classification turn out to be identical. Also in §3 we prove a metrization theorem for locally bounded spaces, which is related to the normal metrization theorem for uniform spaces, but which uses a different induction procedure. In §4 we introduce a large class of linear topological spaces which includes the locally convex spaces and the locally bounded spaces, and for which one of the more important results on boundedness in locally convex spaces is valid. -
1 Definition
Math 125B, Winter 2015. Multidimensional integration: definitions and main theorems 1 Definition A(closed) rectangle R ⊆ Rn is a set of the form R = [a1; b1] × · · · × [an; bn] = f(x1; : : : ; xn): a1 ≤ x1 ≤ b1; : : : ; an ≤ x1 ≤ bng: Here ak ≤ bk, for k = 1; : : : ; n. The (n-dimensional) volume of R is Vol(R) = Voln(R) = (b1 − a1) ··· (bn − an): We say that the rectangle is nondegenerate if Vol(R) > 0. A partition P of R is induced by partition of each of the n intervals in each dimension. We identify the partition with the resulting set of closed subrectangles of R. Assume R ⊆ Rn and f : R ! R is a bounded function. Then we define upper sum of f with respect to a partition P, and upper integral of f on R, X Z U(f; P) = sup f · Vol(B); (U) f = inf U(f; P); P B2P B R and analogously lower sum of f with respect to a partition P, and lower integral of f on R, X Z L(f; P) = inf f · Vol(B); (L) f = sup L(f; P): B P B2P R R R RWe callR f integrable on R if (U) R f = (L) R f and in this case denote the common value by R f = R f(x) dx. As in one-dimensional case, f is integrable on R if and only if Cauchy condition is satisfied. The Cauchy condition states that, for every ϵ > 0, there exists a partition P so that U(f; P) − L(f; P) < ϵ. -
A Representation of Partially Ordered Preferences Teddy Seidenfeld
A Representation of Partially Ordered Preferences Teddy Seidenfeld; Mark J. Schervish; Joseph B. Kadane The Annals of Statistics, Vol. 23, No. 6. (Dec., 1995), pp. 2168-2217. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28199512%2923%3A6%3C2168%3AAROPOP%3E2.0.CO%3B2-C The Annals of Statistics is currently published by Institute of Mathematical Statistics. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ims.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Tue Mar 4 10:44:11 2008 The Annals of Statistics 1995, Vol. -
Compact and Totally Bounded Metric Spaces
Appendix A Compact and Totally Bounded Metric Spaces Definition A.0.1 (Diameter of a set) The diameter of a subset E in a metric space (X, d) is the quantity diam(E) = sup d(x, y). x,y∈E ♦ Definition A.0.2 (Bounded and totally bounded sets)A subset E in a metric space (X, d) is said to be 1. bounded, if there exists a number M such that E ⊆ B(x, M) for every x ∈ E, where B(x, M) is the ball centred at x of radius M. Equivalently, d(x1, x2) ≤ M for every x1, x2 ∈ E and some M ∈ R; > { ,..., } 2. totally bounded, if for any 0 there exists a finite subset x1 xn of X ( , ) = ,..., = such that the (open or closed) balls B xi , i 1 n cover E, i.e. E n ( , ) i=1 B xi . ♦ Clearly a set is bounded if its diameter is finite. We will see later that a bounded set may not be totally bounded. Now we prove the converse is always true. Proposition A.0.1 (Total boundedness implies boundedness) In a metric space (X, d) any totally bounded set E is bounded. Proof Total boundedness means we may choose = 1 and find A ={x1,...,xn} in X such that for every x ∈ E © The Editor(s) (if applicable) and The Author(s), under exclusive 429 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 430 Appendix A: Compact and Totally Bounded Metric Spaces inf d(xi , x) ≤ 1. -
Cantor and Continuity
Cantor and Continuity Akihiro Kanamori May 1, 2018 Georg Cantor (1845-1919), with his seminal work on sets and number, brought forth a new field of inquiry, set theory, and ushered in a way of proceeding in mathematics, one at base infinitary, topological, and combinatorial. While this was the thrust, his work at the beginning was embedded in issues and concerns of real analysis and contributed fundamentally to its 19th Century rigorization, a development turning on limits and continuity. And a continuing engagement with limits and continuity would be very much part of Cantor's mathematical journey, even as dramatically new conceptualizations emerged. Evolutionary accounts of Cantor's work mostly underscore his progressive ascent through set- theoretic constructs to transfinite number, this as the storied beginnings of set theory. In this article, we consider Cantor's work with a steady focus on con- tinuity, putting it first into the context of rigorization and then pursuing the increasingly set-theoretic constructs leading to its further elucidations. Beyond providing a narrative through the historical record about Cantor's progress, we will bring out three aspectual motifs bearing on the history and na- ture of mathematics. First, with Cantor the first mathematician to be engaged with limits and continuity through progressive activity over many years, one can see how incipiently metaphysical conceptualizations can become systemati- cally transmuted through mathematical formulations and results so that one can chart progress on the understanding of concepts. Second, with counterweight put on Cantor's early career, one can see the drive of mathematical necessity pressing through Cantor's work toward extensional mathematics, the increasing objectification of concepts compelled, and compelled only by, his mathematical investigation of aspects of continuity and culminating in the transfinite numbers and set theory. -
Chapter 7. Complete Metric Spaces and Function Spaces
43. Complete Metric Spaces 1 Chapter 7. Complete Metric Spaces and Function Spaces Note. Recall from your Analysis 1 (MATH 4217/5217) class that the real numbers R are a “complete ordered field” (in fact, up to isomorphism there is only one such structure; see my online notes at http://faculty.etsu.edu/gardnerr/4217/ notes/1-3.pdf). The Axiom of Completeness in this setting requires that ev- ery set of real numbers with an upper bound have a least upper bound. But this idea (which dates from the mid 19th century and the work of Richard Dedekind) depends on the ordering of R (as evidenced by the use of the terms “upper” and “least”). In a metric space, there is no such ordering and so the completeness idea (which is fundamental to all of analysis) must be dealt with in an alternate way. Munkres makes a nice comment on page 263 declaring that“completeness is a metric property rather than a topological one.” Section 43. Complete Metric Spaces Note. In this section, we define Cauchy sequences and use them to define complete- ness. The motivation for these ideas comes from the fact that a sequence of real numbers is Cauchy if and only if it is convergent (see my online notes for Analysis 1 [MATH 4217/5217] http://faculty.etsu.edu/gardnerr/4217/notes/2-3.pdf; notice Exercise 2.3.13). 43. Complete Metric Spaces 2 Definition. Let (X, d) be a metric space. A sequence (xn) of points of X is a Cauchy sequence on (X, d) if for all ε > 0 there is N N such that if m, n N ∈ ≥ then d(xn, xm) < ε. -
Topology Proceedings
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. Topology Proceedings Vol 18, 1993 H-BOUNDED SETS DOUGLAS D. MOONEY ABSTRACT. H-bounded sets were introduced by Lam brinos in 1976. Recently they have proven to be useful in the study of extensions of Hausdorff spaces. In this con text the question has arisen: Is every H-bounded set the subset of an H-set? This was originally asked by Lam brinos along with the questions: Is every H-bounded set the subset of a countably H-set? and Is every countably H-bounded set the subset of a countably H-set? In this paper, an example is presented which gives a no answer to each of these questions. Some new characterizations and properties of H-bounded sets are also examined. 1. INTRODUCTION The concept of an H-bounded set was defined by Lambrinos in [1] as a weakening of the notion of a bounded set which he defined in [2]. H-bounded sets have arisen naturally in the au thor's study of extensions of Hausdorff spaces. In [3] it is shown that the notions of S- and O-equivalence defined by Porter and Votaw [6] on the upper-semilattice of H-closed extensions of a Hausdorff space are equivalent if and only if every closed nowhere dense set is an H-bounded set. As a consequence, it is shown that a Hausdorff space has a unique Hausdorff exten sion if and only if the space is non-H-closed, almost H-closed, and every closed nowhere dense set is H-bounded. -
On the Polyhedrality of Closures of Multi-Branch Split Sets and Other Polyhedra with Bounded Max-Facet-Width
On the polyhedrality of closures of multi-branch split sets and other polyhedra with bounded max-facet-width Sanjeeb Dash Oktay G¨unl¨uk IBM Research IBM Research [email protected] [email protected] Diego A. Mor´anR. School of Business, Universidad Adolfo Iba~nez [email protected] August 2, 2016 Abstract For a fixed integer t > 0, we say that a t-branch split set (the union of t split sets) is dominated by another one on a polyhedron P if all cuts for P obtained from the first t-branch split set are implied by cuts obtained from the second one. We prove that given a rational polyhedron P , any arbitrary family of t-branch split sets has a finite subfamily such that each element of the family is dominated on P by an element from the subfamily. The result for t = 1 (i.e., for split sets) was proved by Averkov (2012) extending results in Andersen, Cornu´ejols and Li (2005). Our result implies that the closure of P with respect to any family of t-branch split sets is a polyhedron. We extend this result by replacing split sets with polyhedral sets with bounded max-facet-width as building blocks and show that any family of such sets also has a finite dominating subfamily. This result generalizes a result of Averkov (2012) on bounded max-facet-width polyhedra. 1 Introduction Cutting planes (or cuts, for short) are essential tools for solving general mixed-integer programs (MIPs). Split cuts are an important family of cuts for general MIPs, and special cases of split cuts are effective in practice. -
Introductory Topics in the Mathematical Theory of Continuum Mechanics - R J Knops and R
CONTINUUM MECHANICS - Introductory Topics In The Mathematical Theory Of Continuum Mechanics - R J Knops and R. Quintanilla INTRODUCTORY TOPICS IN THE MATHEMATICAL THEORY OF CONTINUUM MECHANICS R J Knops School of Mathematical and Computer Sciences, Heriot-Watt University, UK R. Quintanilla Department of Applied Mathematics II, UPC Terrassa, Spain Keywords: Classical and non-classical continuum thermomechanics, well- and ill- posedness; non-linear, linear, linearized, initial and boundary value problems, existence, uniqueness, dynamic and spatial stability, continuous data dependence. Contents I. GENERAL PRINCIPLES 1. Introduction 2. The well-posed problem II. BASIC CONTINUUM MECHANICS 3. Introduction 4. Material description 5. Spatial description 6. Constitutive theories III. EXAMPLE: HEAT CONDUCTION 7. Introduction. Existence and uniqueness 8. Continuous dependence. Stability 9. Backward heat equation. Ill-posed problems 10. Non-linear heat conduction 11. Infinite thermal wave speeds IV. CLASSICAL THEORIES 12. Introduction 13. Thermoviscous flow. Navier-Stokes Fluid 14. Linearized and linear elastostatics 15. Linearized Elastodynamics 16. LinearizedUNESCO and linear thermoelastostatics – EOLSS 17. Linearized and linear thermoelastodynamics 18. Viscoelasticity 19. Non-linear Elasticity 20. Non-linearSAMPLE thermoelasticity and thermoviscoelasticity CHAPTERS V. NON-CLASSICAL THEORIES 21. Introduction 22. Isothermal models 23. Thermal Models Glossary Bibliography Biographical sketches ©Encyclopedia of Life Support Systems (EOLSS) CONTINUUM MECHANICS - Introductory Topics In The Mathematical Theory Of Continuum Mechanics - R J Knops and R. Quintanilla Summary Qualitative properties of well-posedness and ill-posedness are examined for problems in the equilibrium and dynamic classical non-linear theories of Navier-Stokes fluid flow and elasticity. These serve as prototypes of more general theories, some of which are also discussed. The article is reasonably self-contained. -
Topology JonPaul Cox Connected Spaces May 11, 2016
Topology JonPaul Cox Connected Spaces May 11, 2016 Connected Spaces JonPaul Cox 5/11/16 6th Period Topology Parallels Between Calculus and Topology Intermediate Value Theorem If is continuous and if r is a real number between f(a) and f(b), then there exists an element such that f(c) = r. Maximum Value Theorem If is continuous, then there exists an element such that for every . Uniform Continuity Theorem If is continuous, then given > 0, there exists > 0 such that for every pair of numbers , of [a,b] for which . 1 Topology JonPaul Cox Connected Spaces May 11, 2016 Parallels Between Calculus and Topology Intermediate Value Theorem If is continuous and if r is a real number between f(a) and f(b), then there exists an element such that f(c) = r. Connected Spaces A space can be "separated" if it can be broken up into two disjoint, open parts. Otherwise it is connected. If the set is not separated, it is connected, and vice versa. 2 Topology JonPaul Cox Connected Spaces May 11, 2016 Connected Spaces Let X be a topological space. A separation of X is a pair of disjoint, nonempty open sets of X whose union is X. E.g. Connected Spaces A space is connected if and only if the only subsets of X that are both open and closed in X are and X itself. 3 Topology JonPaul Cox Connected Spaces May 11, 2016 Subspace Topology Let X be a topological space with topology . If Y is a subset of X (I.E. -
An Introduction to Some Aspects of Functional Analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms on vector spaces, bounded linear operators, and dual spaces of bounded linear functionals in particular. Contents 1 Norms on vector spaces 3 2 Convexity 4 3 Inner product spaces 5 4 A few simple estimates 7 5 Summable functions 8 6 p-Summable functions 9 7 p =2 10 8 Bounded continuous functions 11 9 Integral norms 12 10 Completeness 13 11 Bounded linear mappings 14 12 Continuous extensions 15 13 Bounded linear mappings, 2 15 14 Bounded linear functionals 16 1 15 ℓ1(E)∗ 18 ∗ 16 c0(E) 18 17 H¨older’s inequality 20 18 ℓp(E)∗ 20 19 Separability 22 20 An extension lemma 22 21 The Hahn–Banach theorem 23 22 Complex vector spaces 24 23 The weak∗ topology 24 24 Seminorms 25 25 The weak∗ topology, 2 26 26 The Banach–Alaoglu theorem 27 27 The weak topology 28 28 Seminorms, 2 28 29 Convergent sequences 30 30 Uniform boundedness 31 31 Examples 32 32 An embedding 33 33 Induced mappings 33 34 Continuity of norms 34 35 Infinite series 35 36 Some examples 36 37 Quotient spaces 37 38 Quotients and duality 38 39 Dual mappings 39 2 40 Second duals 39 41 Continuous functions 41 42 Bounded sets 42 43 Hahn–Banach, revisited 43 References 44 1 Norms on vector spaces Let V be a vector space over the real numbers R or the complex numbers C. -
Big Homotopy Theory
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2013 Big Homotopy Theory Keith Gordon Penrod [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Penrod, Keith Gordon, "Big Homotopy Theory. " PhD diss., University of Tennessee, 2013. https://trace.tennessee.edu/utk_graddiss/1768 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Keith Gordon Penrod entitled "Big Homotopy Theory." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Jurek Dydak, Major Professor We have read this dissertation and recommend its acceptance: Jim Conant, Nikolay Brodskiy, Morwen Thistletwaite, Delton Gerloff Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Big Homotopy Theory A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Keith Gordon Penrod May 2013 c by Keith Gordon Penrod, 2013 All Rights Reserved. ii This dissertation is dedicated to Archimedes, who was my inspiration for becoming a mathematician.