DOCSLIB.ORG

1 the Topology of Metric Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 the Topology of Metric Spaces Introductory Analysis I Fall 2014 Notes on Metric Spaces These notes are an alternative to the textbook, from and including "Closed Sets and Open Sets" (page 58) to and excluding Cantor Sets (page 95) 1 The topology of metric spaces Assume M is a metric space with distance function d. I will depart a bit from the book's notation; instead of using Mr(p), I will denote by B(p; r) the open ball of center p 2 M and radius r > 0; i.e, B(p; r) = fq 2 M : d(q; p) < rg: The reason is that once we get to concrete metric spaces like Rn, it may be silly n to write (R )p(r). More importantly, I like my notation better. Incidentally a notation such as "r > 0" means: "r is a positive real number." The basic definition is Definition 1 A subset U of M is said to be open iff for every p 2 U there is r > 0 such that B(p; r) ⊂ U. In other words, the set U is open if and only if for every p 2 U there is r > 0 such that if q 2 U and d(q; p) < r, then q 2 U. Examples: 1. Let M be an arbitrary metric space. Then the empty set ; and M are open. 2. Let M be an arbitrary metric space. Let p 2 M and r > 0. Then B(p; r) is open. 3. Let M be a discrete metric space; i.e., d(p; q) = 1 whenever p =6 q. Then every subset of M is open. 4. Let M = Rm. The following sets are open: m U1 = fx = (x1; : : : ; xm) 2 R : x1 > 0g; m U2 = fx = (x1; : : : ; xm) 2 R : x1 > x2g; m U3 = fx = (x1; : : : ; xm) 2 R : jx1j + ··· + jxmj < 1g; 5. Let M = R. A subset U of R is open if and only if for each x 2 R there exist a; b 2 R such that x 2 (a; b) ⊂ U. In particular, all open intervals are open. You should be able to prove that the sets declared open in these examples are indeed open. 1 THE TOPOLOGY OF METRIC SPACES 2 Theorem 1 Let M be a metric space and let T be the family of all open subsets of M. This family has the following properties: 1. ;;M 2 T . S 2 T 2 2 T 2. If Uα for all α A; A some index set, then α2A Uα . In words, T is closed under arbitrary unions. An equivalent way this property is sometimes formulated is: Assume U is a collection ofS open subsets of M (finite, infinite, countable or non-countable). Then fU : U 2 Ug is open. 3. If U; V 2 T , then U \ V 2 T . The intersection of two open sets is open. By induction, it follows that the intersection of any finite number of open sets is open. Proof. 1. That ;;M are open was already mentioned. S 2 T 2 2 T 2. Let Uα for all α A; A some index set, let U = α2A Uα . Let p 2 U. Then there exists α 2 A such that p 2 Uα. Since Uα is open, there is r > 0 such that B(p; r) ⊂ Uα. Since Uα ⊂ U, B(p; r) ⊂ U. 3. Let U; V be open. If p 2 U \ V , then p 2 U and p 2 V . Since U is open, p 2 U implies that there exists r1 > 0 such that B(p; r1) ⊂ U. Similarly, because V is open, there exists r2 > 0 such that B(p; r2) ⊂ V . Let r = min(r1; r2). Then r > 0 and clearly B(p; r) ⊂ B(p; r1) \ B(p; r2) ⊂ U \ V . Exercise 1 Let M = R. 1. Prove that in R singleton sets are NOT open. 2 N f 2 R − g 2. For n let Un =T x : 1=n < x < 1=n . Prove that Un is open 2 N 1 for all n , but n=1 Un is not open. This shows that, in general, the intersection of an infinite family of open sets may not be open. Note: If M is any set, a family of subsets of M satisfying properties 1, 2, and 3 of Theorem 1 is called a topology for M. Definition 2 Let M be a metric space. A subset F of M is closed iff its complement MnF is open. Examples: 1 THE TOPOLOGY OF METRIC SPACES 3 1. Let M be an arbitrary metric space. Then the empty set ; and M are closed. 2. Let M be an arbitrary metric space. Let p 2 M and r ≥ 0. Then the closed ball of center p, radius r; that is, the set fq 2 M : d(q; p) ≤ rg is closed. 3. A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. 4. Let M be a discrete metric space; i.e., d(p; q) = 1 whenever p =6 q. Then every subset of M is closed. 5. Let M = Rm. The following sets are closed: m U1 = fx = (x1; : : : ; xm) 2 R : x1 ≥ 0g; m U2 = fx = (x1; : : : ; xm) 2 R : x1 ≥ x2g; m U3 = fx = (x1; : : : ; xm) 2 R : jx1j + ··· + jxmj ≤ 11g; 6. Let M = R. All closed intervals are closed. As mentioned in the examples, in R open intervals are open and closed intervals are closed. Intervals of the form (a; b]; [a; b) with a; b 2 R, a < b are neither open nor closed. By taking complements, the following theorem is an immediate consequence of Theorem 1 Theorem 2 Let M be a metric space. 1. ;;M 2 T are closed. 2 2. IfT Fα is a closed subset of M for all α A; A some index set, then α2A Uα is closed. In words, arbitrary intersections of closed subsets are closed. Equivalently: Assume F is a collection ofT closed subsets of M (finite, infinite, countable or non-countable). Then fF : F 2 Fg is closed. 3. If F; G, are closed then F [ G is closed. By induction, it follows that the Union of any finite number of closed sets is closed. Our next task will be to show that the basic metric properties can be de- scribed exclusively in terms of open sets. Another way of stating this is to say that one never has to mention the distance function again, except if one wants to. It will be convenient to first have another topological concept, that of neighborhood. Definition 3 Let M be a metric space. Let p 2 M. A subset V of M is said to be a neighborhood of p in M (or just a neighborhood of p) iff there exists an open set U such that p 2 U ⊂ V . 1 THE TOPOLOGY OF METRIC SPACES 4 Exercise 2 V is a neighborhood of p if and only if there exists r > 0 such that B(p; r) ⊂ V . We also have the following simple lemma Lemma 3 A subset U of a metric space is open if and only if it is a neighbor- hood of each of its points. Proof. Assume first U is open and let p 2 U. Then p 2 U ⊂ U, showing U is a neighborhood of p. Conversely, assume U is a neighborhood of eachS of its points. 2 2 ⊂ Then for each p U, there is Wp open, p Wp U. Let W = p2U Wp. By property 2 of Theorem 1, W is open. Since Wp ⊂ U for all p 2 U, we have W ⊂ U. On the other hand, if p 2 U, then p 2 Wp ⊂ W , thus U ⊂ W . It follows that U = W is open. Let us begin with sequences. We'll state it in the form of a theorem, but it is fairly immediate. Theorem 4 Let (pn) be a sequence in the metric space M; let p 2 M. The following statements are equivalent: 1. For each ϵ > 0 there exists N 2 N such that n ≥ N implies d(pn; p) < ϵ. 2. For each ϵ > 0 there exists N 2 R such that n > N implies d(pn; p) < ϵ. 3. For each open set U in M such that p 2 U, there exists N 2 R such that n > N implies pn 2 U. 4. For each neighborhood V of p, there exists N 2 R such that n > N implies pn 2 V . 5. If V is a neighborhood of p, then fn 2 N : pn 2= V g is finite. As mentioned before, most of the equivalences are obvious. In lieu of a proof I'll just make some comments about what could be the less obvious parts. Concerning the equivalence of 1 and 2, notice first that if the following statement "For each there exists N 2 N such that n ≥ N implies " (fill the blanks any way you wish), is equivalent to "For each there exists N 2 N such that n > N implies " In fact, if something holds for n ≥ N, it also holds for n > N. But the converse is also true; if we can find N 2 N so that a property holds for n > N, we can also find N 0 so it holds for n ≥ N 0; namely N 0 = N + 1. Maybe in words: if there is an element of N so the property holds when n is strictly larger than that element of N, then there is an element of N (namely the previous element plus 1) such that the property holds for n greater than or equal this element.
Load more

Recommended publications
  • Metric Spaces 2000, Lecture 17
    Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f −1 is not continuous. For another example, consider the sets T = [0, 2π) ⊆ R and S = { (x, y) ∈ R2 | x2 + y2 = 1 }, and let f: T → S be given by f(θ) = (cos θ, sin θ). It is trivial that f is bijective. Viewed geometrically, f wraps the interval T around the circle S, joining the ends together. We consider T as a subspace of R and S as a subspace of R2, using the usual (Euclidean) metrics on R and R2, so that the concept of continuity for functions between S and T is the familiar concept studied in calculus courses. The function f is thus clearly continuous, since its component functions, cos and sin, are differentiable and therefore continuous. Joining the ends of the interval together is thus perfectly compatible with continuity. However, the inverse function g: S → T cuts the circle open at the point (1, 0) and unwraps it. This is not a continuous operation: some points that are close together in S are mapped by g to points that are far apart in T . B A A0 B0 A00 π π 022 g S T In the diagram, A is the point (1, 0) in S, and g maps it to the point A0 in T . But points arbitrarily near to A that are below the x-axis are mapped to points in T that are near to A00 rather than A0.
    [Show full text]
  • Metric Spaces
    Chapter 1. Metric Spaces Definitions. A metric on a set M is a function d : M M R × → such that for all x, y, z M, Metric Spaces ∈ d(x, y) 0; and d(x, y)=0 if and only if x = y (d is positive) MA222 • ≥ d(x, y)=d(y, x) (d is symmetric) • d(x, z) d(x, y)+d(y, z) (d satisfies the triangle inequality) • ≤ David Preiss The pair (M, d) is called a metric space. [email protected] If there is no danger of confusion we speak about the metric space M and, if necessary, denote the distance by, for example, dM . The open ball centred at a M with radius r is the set Warwick University, Spring 2008/2009 ∈ B(a, r)= x M : d(x, a) < r { ∈ } the closed ball centred at a M with radius r is ∈ x M : d(x, a) r . { ∈ ≤ } A subset S of a metric space M is bounded if there are a M and ∈ r (0, ) so that S B(a, r). ∈ ∞ ⊂ MA222 – 2008/2009 – page 1.1 Normed linear spaces Examples Definition. A norm on a linear (vector) space V (over real or Example (Euclidean n spaces). Rn (or Cn) with the norm complex numbers) is a function : V R such that for all · → n n , x y V , x = x 2 so with metric d(x, y)= x y 2 ∈ | i | | i − i | x 0; and x = 0 if and only if x = 0(positive) i=1 i=1 • ≥ cx = c x for every c R (or c C)(homogeneous) • | | ∈ ∈ n n x + y x + y (satisfies the triangle inequality) Example (n spaces with p norm, p 1).
    [Show full text]
  • Euclidean Space Rn
    Tel Aviv University, 2014/15 Analysis-III,IV 1 1 Euclidean space Rn 1a Prerequisites . 1 n 1b Structures on R and their isomorphisms . 5 1c Metric and topology . 20 1d Linearity and continuity . 25 1e Norms of vectors and operators . 26 The n-dimensional space Rn may be treated as a Euclidean space, or just a vector space, etc. Its topology is uniquely determined by algebraic structure. 1a Prerequisites linear algebra You should know the notion of: Forgot? Then see: Vector space (=linear space) [Sh:p.26 \Vector space axioms"] Isomorphism of vector spaces: a linear bijection. Basis of a vector space [Sh:Def.2.1.2 on p.28] Dimension of a finite-dimensional vector space: the number of vectors in every basis. Two finite-dimensional vector spaces are isomorphic if and only if their di- mensions are equal. Subspace of a vector space. Linear operator (=mapping=function) between vector spaces [Sh:3.1] Inner product on a vector space: hx; yi [Sh:p.31 \Inner product properties"] A basis of a subspace, being a linearly independent system, can be extended to a basis of the whole finite-dimensional vector space. topology You should know the notion of: Forgot? Then see: A sequence of points of Rn [Sh:p.36]1 Its convergence, limit [Sh:p.42{43] 1Quote: The only obstacle . is notation . n already denotes the dimension of the Euclidean space where we are working; and furthermore, the vectors can't be denoted with subscripts since a subscript denotes a component of an individual vector. As our work with vectors becomes more intrinsic, vector entries will demand less of our attention, and Tel Aviv University, 2014/15 Analysis-III,IV 2 Mapping Rn ! Rm; continuity (at a point; on a set) [Sh:p.41{48] Subsequence; Bolzano-Weierstrass theorem [Sh:p.52{53] Subset of Rn, its limit points; closed set; bounded set [Sh:p.51] Compact set [Sh:p.54] Open set [Sh:p.191]1 Closure, boundary, interior [Sh:p.311,314] Open cover; Heine-Borel theorem [Sh:p.312] Open ball, closed ball, sphere [Sh:p.50,191{192] Open box, closed box [Sh:p.246] [Sh:Exer.
    [Show full text]
  • Real Analysis
    Real Analysis William G. Faris May 1, 2006 ii Contents Preface xi I Sets and Functions 1 1 Logical language and mathematical proof 3 1.1 Terms, predicates and atomic formulas . 3 1.2 Formulas . 4 1.3 Restricted variables . 5 1.4 Free and bound variables . 5 1.5 Quantifier logic . 6 1.6 Natural deduction . 8 1.7 Rules for logical operations . 9 1.8 Additional rules for or and exists . 11 1.9 Strategies for natural deduction . 12 1.10 Lemmas and theorems . 14 1.11 Relaxed natural deduction . 15 1.12 Supplement: Templates . 16 1.13 Supplement: Existential hypotheses . 18 2 Sets 23 2.1 Zermelo axioms . 23 2.2 Comments on the axioms . 24 2.3 Ordered pairs and Cartesian product . 27 2.4 Relations and functions . 28 2.5 Number systems . 29 2.6 The extended real number system . 30 2.7 Supplement: Construction of number systems . 31 3 Relations, functions, dynamical Systems 35 3.1 Identity, composition, inverse, intersection . 35 3.2 Picturing relations . 36 3.3 Equivalence relations . 36 3.4 Generating relations . 36 iii iv CONTENTS 3.5 Ordered sets . 37 3.6 Functions . 37 3.7 Relations inverse to functions . 38 3.8 Dynamical systems . 38 3.9 Picturing dynamical systems . 38 3.10 Structure of dynamical systems . 39 3.11 Isomorphism of dynamical systems . 41 4 Functions, cardinal number 43 4.1 Functions . 43 4.2 Picturing functions . 44 4.3 Indexed sums and products . 44 4.4 Cartesian powers . 45 4.5 Cardinality . 45 II Order and Metric 49 5 Ordered sets and order completeness 51 5.1 Ordered sets .
    [Show full text]
  • Arxiv:1903.05504V3 [Math.FA] 22 Apr 2020 2010 Mathematics Subject Classification
    AMALGAMATION AND RAMSEY PROPERTIES OF Lp SPACES V. FERENCZI, J. LOPEZ-ABAD, B. MBOMBO, AND S. TODORCEVIC Abstract. We study the dynamics of the group of isometries of Lp-spaces. In particular, we study the canonical actions of these groups on the space of δ-isometric embeddings of finite dimensional subspaces of Lpp0, 1q into itself, and we show that for every real number 1 ¤ p ă 8 with p ‰ 4, 6, 8,... they are ε-transitive provided that δ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces Lpp0, 1q, p ‰ 4, 6, 8,... and the n Gurarij space. We also give a proof of the Ramsey property of the classes t`p un, p ‰ 2, 8, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X. 1. Introduction It is a classical result of A. Pełczyński and S. Rolewicz [PelRol] that the spaces Lpp0, 1q are almost transitive, in the sense that the group of linear isometric surjections IsopLpp0, 1qq acts almost transitively on the corresponding unit sphere of Lpp0, 1q.
    [Show full text]
  • Real Analysis: Part I
    Real Analysis: Part I William G. Faris February 2, 2004 ii Contents 1 Mathematical proof 1 1.1 Logical language . 1 1.2 Free and bound variables . 3 1.3 Proofs from analysis . 4 1.4 Natural deduction . 6 1.5 Natural deduction strategies . 13 1.6 Equality . 16 1.7 Lemmas and theorems . 18 1.8 More proofs from analysis . 19 2 Sets 21 2.1 Zermelo axioms . 21 2.2 Comments on the axioms . 22 2.3 Ordered pairs and Cartesian product . 25 2.4 Relations and functions . 26 2.5 Number systems . 28 3 Relations, Functions, Dynamical Systems 31 3.1 Identity, composition, inverse, intersection . 31 3.2 Picturing relations . 32 3.3 Equivalence relations . 32 3.4 Generating relations . 32 3.5 Ordered sets . 33 3.6 Functions . 33 3.7 Relations inverse to functions . 34 3.8 Dynamical systems . 34 3.9 Picturing dynamical systems . 35 3.10 Structure of dynamical systems . 35 4 Functions, Cardinal Number 39 4.1 Functions . 39 4.2 Picturing functions . 40 4.3 Indexed sums and products . 40 4.4 Cartesian powers . 41 iii iv CONTENTS 4.5 Cardinality . 41 5 Ordered sets and completeness 45 5.1 Ordered sets . 45 5.2 Order completeness . 46 5.3 Sequences in a complete lattice . 47 5.4 Order completion . 48 5.5 The Knaster-Tarski fixed point theorem . 49 5.6 The extended real number system . 49 6 Metric spaces 51 6.1 Metric space notions . 51 6.2 Normed vector spaces . 51 6.3 Spaces of finite sequences . 52 6.4 Spaces of infinite sequences .
    [Show full text]
  • Some Metrics on Teichm¨Uller Spaces of Surfaces Of
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 8, August 2011, Pages 4109–4134 S 0002-9947(2011)05090-7 Article electronically published on March 23, 2011 SOME METRICS ON TEICHMULLER¨ SPACES OF SURFACES OF INFINITE TYPE LIXIN LIU AND ATHANASE PAPADOPOULOS Abstract. Unlike the case of surfaces of topologically finite type, there are several different Teichm¨uller spaces that are associated to a surface of topolog- ically infinite type. These Teichm¨uller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or confor- mal), on the choice of a distance function on Teichm¨uller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichm¨uller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichm¨uller theory of surfaces of infinite topological type that do not appear in the setting of the Teichm¨uller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichm¨uller spaces associated to a given surface of topologically infinite type. 1. Introduction Let X be a connected oriented surface of infinite topological type. Unlike the case of a surface of finite type, there are several Teichm¨uller spaces associated to X. Such a Teichm¨uller space can be defined as a space of equivalence classes of marked hyperbolic surfaces homeomorphic to X, the marking being a homeomorphism be- tween a base hyperbolic structure on X and a hyperbolic surface homeomorphic to X.
    [Show full text]
  • NASA Scientific and Technical Aerospace Reports
    NASA STI Program ... in Profile Since its founding, NASA has been dedicated • CONFERENCE PUBLICATION. to the advancement of aeronautics and space Collected papers from scientific and science. The NASA scientific and technical technical conferences, symposia, information (STI) program plays a key part in seminars, or other meetings sponsored helping NASA maintain this important role. or co-sponsored by NASA. The NASA STI program operates under the • SPECIAL PUBLICATION. Scientific, auspices of the Agency Chief Information technical, or historical information from Officer. It collects, organizes, provides for NASA programs, projects, and missions, archiving, and disseminates NASA’s STI. The often concerned with subjects having NASA STI program provides access to the substantial public interest. NASA Aeronautics and Space Database and its public interface, the NASA Technical Report • TECHNICAL TRANSLATION. Server, thus providing one of the largest English-language translations of foreign collections of aeronautical and space science scientific and technical material pertinent to STI in the world. Results are published in both NASA’s mission. non-NASA channels and by NASA in the NASA STI Report Series, which includes the Specialized services also include creating following report types: custom thesauri, building customized databases, and organizing and publishing research results. • TECHNICAL PUBLICATION. Reports of completed research or a major significant For more information about the NASA STI phase of research that present the results of program, see the following: NASA Programs and include extensive data or theoretical analysis. Includes • Access the NASA STI program home page compilations of significant scientific and at http://www.sti.nasa.gov technical data and information deemed to be of continuing reference value.
    [Show full text]
  • Review of Metric Spaces
    Lecture 2: Review of Metric Spaces Hart Smith Department of Mathematics University of Washington, Seattle Math 524, Autumn 2013 Hart Smith Math 524 Definition of a Metric Space A metric space consists of: a set X , and function (metric) ρ : X × X ! [0; 1) ; such that: ρ(x; y) = ρ(y; x) (Symmetry) ρ(x; y) = 0 iff x = y (Non-degeneracy) ρ(x; y) + ρ(y; z) ≤ ρ(x; z) (Triangle inequality) Examples of metrics on Rn: n 1=2 X 2 Euclidean metric: ρ(x; y) = jxj − yj j j=1 Box metric: ρ(x; y) = max jxj − yj j j Hart Smith Math 524 Open sets in a metric space A subset O ⊂ X is open if: for each x 2 O there exists δ > 0 ( δ can depend on x ) such that y 2 O whenever ρ(x; y) < δ : B(z; r) ≡ x : ρ(x; z) < r g is open, by triangle inequality. The sets X and ; are both open. The union of any collection of open sets is open. The intersection of a finite collection of open sets is open. The collection of open subsets of X is a topology on X. Hart Smith Math 524 Closed sets in a metric space A subset F ⊂ X is closed if: the complement F c ≡ XnF is open. The intersection of any collection of closed sets is closed. The union of a finite collection of closed sets is closed. For any set E ⊂ X, define the interior and the closure of E: Interior = largest open set contained in E: Eo = int(E) = [ O : O ⊂ E is open Closure = smallest closed set containing E: E = \ F : F ⊃ E is closed Hart Smith Math 524 Sequences in a metric space 1 A sequence fxng ⊂ X converges to x if lim ρ(xn; x) = 0 : n=1 n!1 1 A sequence fxng ⊂ X is Cauchy if lim ρ(xm; xn) = 0 : n=1 m;n!1 8 > 0 ; 9 N < 1 ; such that ρ(xm; xn) < if m; n > N : Every convergent sequence is Cauchy 1 The point x is a cluster point of the sequence fxngn=1 if, for every r > 0, B(x; r) contains xn for infinitely many n.
    [Show full text]
  • Mathematics for Control Theory
    Mathematics for Control Theory Elements of Analysis p.1 Metric Spaces Contraction Mapping Principle Hanz Richter Mechanical Engineering Department Cleveland State University Reading materials We will use: ■ Anthony N. Michel and Charles J. Herget (2007) [1981], Algebra and Analysis for Engineers and Scientists, Birkhäuser, e-ISBN-13: 978-0-8176-4707-0 (chapter 5, skip sections indicated by the instructor) ■ Kenneth R. Davidson and Allan P. Donsig [2002], Real Analysis with Real Applications, Prentice-Hall, ISBN 0-13-041647-9 (use chapter 4 as a reference). 2 / 30 Introduction Metric spaces are the most general setting in which we can study important aspects of sequences and functions such as convergence and continuity. Essential concepts pertaining to the base sets of metric spaces, such as openness and compactness and their relation to functions defined on them will be covered in this section. Metric spaces are sets where a distance function (the metric) is defined for any pair of its elements. Only a few requirements are made for the metric; otherwise, the definition is very general. 3 / 30 Definition of metric space Let X be any nonempty set and let ρ be a real-valued function defined on X X satisfying: × 1. ρ(x, y) 0 x, y X and ρ(x, y)=0 x = y. ≥ ∀ ∈ ⇐⇒ 2. ρ(x, y) = ρ(y,x) x, y X ∀ ∈ 3. ρ(x, y) ρ(x,z) + ρ(z, y) x, y X ≤ ∀ ∈ Metric spaces defined as above will be denoted X; ρ . A given X admits { } infinitely many metrics. Examples: ρ(x, y) = x y is a metric when X = R.
    [Show full text]
  • A Class of New Bi-Invariant Metrics on the Hamiltonian Diffeomorphism
    A class of new bi-invariant metrics on the Hamiltonian diffeomorphism groups Guangcun Lu ∗ Tie Sun School of Mathematical Sciences, Beijing Normal University Laboratory of Mathematics and Complex Systems, Ministry of Education Beijing 100875, The People’s Republic of China ([email protected] [email protected]) May 14, 2018 Abstract In this paper, we construct infinitely many bi-invariant metrics on the Hamiltonian diffeomorphism group and study their basic properties and corresponding generaliza- tions of the Hofer inequality and Sikorav one. Key words: Hamiltonian diffeomorphism, Hofer’s metric, bi-invariant Finsler metrics arXiv:1406.5878v1 [math.SG] 23 Jun 2014 1 Introduction and Main Results 1.1 The Hofer metric In 1989, H. Hofer [12] constructed a remarkable bi-invariant Finsler metric on the group of compactly supported Hamiltonian diffeomorphisms Ham(M,ω) of a sym- plectic manifold (M,ω), nowadays known as Hofer metric. Since then the intrinsic geometry of it has been being a very active and fruitful research field in symplec- tic topology and Hamiltonian dynamics (see the books [14, 18, 27], and the surveys [10, 19, 28, 24] and references therein for current progress situation). Especially, a recent celebrated result made by Buhovsky and Ostrover [5] is a positive answer to the uniqueness question of the Hofer metric raised by Eliashberg ∗2010 Mathematics Subject Classification. Primary 53D22, 53D40, 53D25. Partially supported by the NNSF 10971014 and 11271044 of China, PCSIRT, RFDPHEC (No. 200800270003) and the Fundamental Research Funds for the Central Universities (No. 2012CXQT09). 1 and Polterovich [9]. They showed that up to equivalence of metrics the Hofer metric is the only bi-invariant Finsler metric on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold under a natural assumption.
    [Show full text]
  • Finite Metric Spaces and Their Embedding Into Lebesgue Spaces
    FINITE METRIC SPACES AND THEIR EMBEDDING INTO LEBESGUE SPACES RYAN HOPKINS Abstract. The properties of the metric topology on infinite and finite sets are analyzed. We answer whether finite metric spaces hold interest in algebraic topology, and how this result is generalized to pseudometric spaces through the Kolmogorov quotient. Embedding into Lebesgue spaces is analyzed, with special attention for Hilbert spaces, `p, and EN . Contents 1. Introduction 1 2. Finite Metric Spaces 2 2.1. Pseudometrizing Finite Spaces 2 2.2. Representing Metrics on Finite Spaces 3 3. The Problem with Finite Metric Spaces 4 3.1. The Discrete Topology 4 3.2. The Kolmogorov Quotient 4 4. Embedding Finite Metric Spaces 6 4.1. Embedding in `2 8 4.2. Embedding in `1 11 4.3. Embedding in `1 11 4.4. Embedding in RN 12 4.5. Embeddings of the `2 Metric 18 Acknowledgments 19 References 19 1. Introduction Finite metric spaces are simple objects, a finite collection of points with a real distance defined between each pair. Despite their apparent simplicity, they are intriguing. From the perspective of algebraic topology, they have no interest as dis- crete spaces. Although relaxing metrics to pseudometrics appears to provide finite metric spaces with more interest, pseudometric spaces are homotopically equiva- lent to the discrete space formed when they are passed through the Kolmogorov quotient. Despite their uninteresting topogical structure, finite metric spaces have applications to computer science. Many physical systems can be modeled with fi- nite points and distances between them, so computer scientists are motivated to embed finite metric spaces into host spaces like RN where detailed analysis can be Date: August 17 2015.
    [Show full text]