DOCSLIB.ORG

Magnitude Homology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Magnitude Homology Magnitude homology Tom Leinster Edinburgh A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: Plan 1. The idea of magnitude 2. The magnitude of a metric space 3. The idea of magnitude homology 4. The magnitude homology of a metric space 1. The idea of magnitude Size For many types of mathematical object, there is a canonical notion of size. • Sets have cardinality. It satisfies jX [ Y j = jX j + jY j − jX \ Y j jX × Y j = jX j × jY j : n • Subsets of R have volume. It satisfies vol(X [ Y ) = vol(X ) + vol(Y ) − vol(X \ Y ) vol(X × Y ) = vol(X ) × vol(Y ): • Topological spaces have Euler characteristic. It satisfies χ(X [ Y ) = χ(X ) + χ(Y ) − χ(X \ Y ) (under hypotheses) χ(X × Y ) = χ(X ) × χ(Y ): Challenge Find a general definition of `size', including these and other examples. One answer The magnitude of an enriched category. Enriched categories A monoidal category is a category V equipped with a product operation. A category X enriched in V is like an ordinary category, but each HomX(X ; Y ) is now an object of V (instead of a set). linear categories metric spaces categories enriched posets categories monoidal • Vect • Set • ([0; 1]; ≥) categories • (0 ! 1) The magnitude of an enriched category There is a general definition of the magnitude jXj of an enriched category. (Definition and details omitted.) Examples This gives definitions of: • the magnitude of a poset • the magnitude of an ordinary category • the magnitude of a linear category • the magnitude of a metric space. The magnitude of a poset The magnitude of a poset is better known as its Euler characteristic (1960s). Example Let M be a triangulated manifold. Write P for the poset of simplices in the triangulation, ordered by inclusion. Then jPj = χ(M): The magnitude of an ordinary category Let X be a finite category. `Recall': X gives rise to a topological space BX (its classifying space), built as follows: • for each object of X, put a point • into BX; • for each map x ! y in X, put an interval •||• into BX; • for each commutative triangle in X, put a 2-simplex N into BX; • ... Theorem Let X be a finite category. Then jXj = χ(BX); under hypotheses ensuring that χ(BX) is well-defined. The magnitude of a linear category . is related to the Euler form in commutative algebra. The magnitude of a metric space . is something new! 2. The magnitude of a metric space | done explicitly | The magnitude of a finite metric space Let X be a finite metric space. Write ZX for the X × X matrix with entries −d(x;y) ZX (x; y) = e (x; y 2 X ). (Why e−distance? Because e−(u+v) = e−ue−v .) n If ZX is invertible (which it is if X ⊆ R ), the magnitude of X is X −1 jX j = ZX (x; y) 2 R x;y2X |the sum of all the entries of the inverse matrix of ZX . First examples • j;j = 0. • |•| = 1. −0 −`−1 ` ! e e 2 • • • = sum of entries of = e−` e−0 1 + e−` 2 1 0 ` • If d(x; y) = 1 for all x 6= y then jX j = cardinality(X ). Slogan: Magnitude is the ‘effective number of points'. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space. Example: a 3-point space (Simon Willerton) Take the 3-point space X = • When t is small, X looks like a 1-point space. • • When t is large, X looks like a 3-point space. Example: a 3-point space (Simon Willerton) Take the 3-point space X = • When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space. •• Example: a 3-point space (Simon Willerton) Take the 3-point space X = • When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space. • • • Example: a 3-point space (Simon Willerton) Take the 3-point space X = • When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space. Indeed, the magnitude of X as a function of t is: Magnitude functions Magnitude assigns to each metric space not just a number, but a function. For t > 0, write tX for X scaled up by a factor of t. The magnitude function of a metric space X is the partially-defined function (0; 1) ! R t 7! jtX j : E.g.: the magnitude function of X = (• ` !•) is jtX j 2 2=(1 + e−`t ) 1 0 t n A magnitude function has only finitely many singularities (none if X ⊆ R ). It is increasing for t 0, and lim jtX j = cardinality(X ). t!1 • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again. Dimension at different scales 2 Let X = , with subspace metric from R . • When t is small, tX looks 0-dimensional. • • When t is large, tX looks 0-dimensional again. Dimension at different scales 2 Let X = , with subspace metric from R . • When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. Dimension at different scales • 2 Let X = , with subspace metric from R . • When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again. • • Dimension at different scales 2 Let X = , with subspace metric from R . • When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again. The magnitude function sees all this! Here's how. Dimension at different scales (Willerton) For a function f : (0; 1) ! R, the instantaneous growth of f at t 2 (0; 1) is d(log f (t)) growth(f ; t)= = slope of the log-log graph of f at t: d(log t) E.g.: If f (t) = Ctn then growth(f ; t) = n for all t. For a space X , the magnitude dimension of X at scale t is dim(X ; t) = growth(jtX j ; t): 1:0 E.g.: Let X = . X dim(X ; t) 0:5 0 0 10 20 30 40 50 t The magnitude of a compact metric space A metric space M is positive definite if for every finite Y ⊆ M, the matrix ZY is positive definite. n E.g.: R with Euclidean or taxicab metric; sphere with geodesic metric; hyperbolic space; any ultrametric space. Theorem (Mark Meckes) All sensible ways of extending the definition of magnitude from finite metric spaces to compact positive definite spaces are equivalent. For a compact positive definite space X , jX j = supfjY j : finite Y ⊆ X g: Magnitude encodes geometric information Theorem (Juan-Antonio Barcel´o& Tony Carbery) n For compact X ⊆ R , jtX j voln(X ) = Cn lim t!1 tn where Cn is a known constant. Theorem (Heiko Gimperlein & Magnus Goffeng) n Assume n is odd. For `nice' compact X ⊆ R (meaning that @X is smooth and Cl(Int(X )) = X ), n n−1 n−2 jtX j = cn voln(X )t + cn−1 voln−1(@X )t + O(t ) as t ! 1, where cn and cn−1 are known constants. The magnitude function knows the volume and the surface area. Magnitude encodes geometric information Magnitude satisfies an asymptotic inclusion-exclusion principle: Theorem (Gimperlein & Goffeng) n Assume n is odd. Let X ; Y ⊆ R with X , Y and X \ Y nice. Then jt(X [ Y )j + jt(X \ Y )j − jtX j − jtY j ! 0 as t ! 1. n n But not all results are asymptotic! Let B denote the unit ball in R . Theorem (Barcel´o& Carbery; Willerton) n Assume n is odd. Then jtB j is a known rational function of t over Z. Examples • tB1 = j[−t; t]j = 1 + t 3 2 1 3 • tB = 1 + 2t + t + 3! t 2 3 4 5 5 24 + 72t + 35t + 9t + t t • tB5 = + . 8(3 + t) 5! 3. The idea of magnitude homology Two perspectives on Euler characteristic So far: Euler characteristic has been treated as an analogue of cardinality. Alternatively: Given any homology theory H∗ of any kind of object X , can define 1 X n χ(X )= (−1) rank Hn(X ): n=0 Note: • χ(X ) is a number • H∗(X ) is an algebraic structure, and functorial in X . We say that H∗ is a categorification of χ. So, homology categorifies Euler characteristic. Towards magnitude homology of enriched categories Any ordinary category X gives rise to a chain complex C∗(X): a Cn(X)= Z · X(x0; x1) × · · · × X(xn−1; xn) x0;:::;xn2X where Z · −: Set ! Ab is the free abelian group functor. The homology H∗(X) of X is the homology of C∗(X). Theorem H∗(X) = H∗(BX). Generalizing this, Michael Shulman gave a definition of the homology of an enriched category (omitted here).
Load more

Recommended publications
  • Metric Geometry in a Tame Setting
    University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures .
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Tennessee, Knoxville: Trace University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2011 Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected] Recommended Citation Wilkins, Leonard Duane, "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces. " PhD diss., University of Tennessee, 2011. https://trace.tennessee.edu/utk_graddiss/1039 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 Leonard Duane Wilkins entitled "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." 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 requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Conrad P. Plaut, Major Professor We have read this dissertation and recommend its acceptance: James Conant, Fernando Schwartz, Michael Guidry Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled \Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." 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 requirements for the degree of Doctor of Philosophy, with a major in Mathematics.
    [Show full text]
  • Notes on Metric Spaces
    Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Still, you should check the corresponding sections in the book for a possibly different point of view on a few things. The main idea to have in mind is that a metric space is some kind of generalization of R in the sense that it is some kind of \space" which has a notion of \distance". Having such a \distance" function will allow us to phrase many concepts from real analysis|such as the notions of convergence and continuity|in a more general setting, which (somewhat) surprisingly makes many things actually easier to understand. Metric Spaces Definition 1. A metric on a set X is a function d : X × X ! R such that • d(x; y) ≥ 0 for all x; y 2 X; moreover, d(x; y) = 0 if and only if x = y, • d(x; y) = d(y; x) for all x; y 2 X, and • d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X. A metric space is a set X together with a metric d on it, and we will use the notation (X; d) for a metric space. Often, if the metric d is clear from context, we will simply denote the metric space (X; d) by X itself.
    [Show full text]
  • Math 118: Topology in Metric Spaces
    Math 118: Topology in Metric Spaces You may recall having seen definitions of open and closed sets, as well as other topological concepts such as limit and isolated points and density, for sets of real numbers (or even sets of points in Euclidean space). A lot of these concepts help us define in analysis what we mean by points being “close” to one another so that we can discuss convergence and continuity. We are going to re-examine all of these concepts for subsets of more general spaces: One on which we have a nicely defined distance. Definition A metric space (X, d) is a set X, whose elements we call points, along with a function d : X × X → R that satisfies (a) (b) (c) For any two points p, q ∈ X, d(p, q) is called the distance from p to q. Notes: We clearly want non-negative distances, as well as symmetry. If we think of points in the Euclidean space R2, or the complex plane, property (c) is the usual triangle inequality. Also note that for any Y ⊆ X, (Y, d) is a metric space. Examples You may have noticed that in a lot of these examples, the distance is of a special form. Consider a normed vector space (a vector space, as you learned in linear algebra, allows for the addition of the elements, as well as multiplication by a scalar. A norm is a real-valued function on the vector space that satisfies the triangle inequality, is 0 only for the 0 vector, and multiplying a vector by a scalar α just increases its norm by the factor |α|.) The norm always allows us to define a distance: for two elements v, w in the space, d(v, w) = kv −wk.
    [Show full text]
  • Chapter 2 Metric Spaces and Topology
    2.1. METRIC SPACES 29 Definition 2.1.29. The function f is called uniformly continuous if it is continu- ous and, for all > 0, the δ > 0 can be chosen independently of x0. In precise mathematical notation, one has ( > 0)( δ > 0)( x X) ∀ ∃ ∀ 0 ∈ ( x x0 X d (x , x0) < δ ), d (f(x ), f(x)) < . ∀ ∈ { ∈ | X 0 } Y 0 Definition 2.1.30. A function f : X Y is called Lipschitz continuous on A X → ⊆ if there is a constant L R such that dY (f(x), f(y)) LdX (x, y) for all x, y A. ∈ ≤ ∈ Let fA denote the restriction of f to A X defined by fA : A Y with ⊆ → f (x) = f(x) for all x A. It is easy to verify that, if f is Lipschitz continuous on A ∈ A, then fA is uniformly continuous. Problem 2.1.31. Let (X, d) be a metric space and define f : X R by f(x) = → d(x, x ) for some fixed x X. Show that f is Lipschitz continuous with L = 1. 0 0 ∈ 2.1.3 Completeness Suppose (X, d) is a metric space. From Definition 2.1.8, we know that a sequence x , x ,... of points in X converges to x X if, for every δ > 0, there exists an 1 2 ∈ integer N such that d(x , x) < δ for all i N. i ≥ 1 n = 2 n = 4 0.8 n = 8 ) 0.6 t ( n f 0.4 0.2 0 1 0.5 0 0.5 1 − − t Figure 2.1: The sequence of continuous functions in Example 2.1.32 satisfies the Cauchy criterion.
    [Show full text]
  • Metric Foundations of Geometry. I
    METRIC FOUNDATIONS OF GEOMETRY. I BY GARRETT BIRKHOFF 1. Introduction. It is shown below by elementary methods that «-dimen- sional Euclidean, spherical, and hyperbolic geometry can be characterized by the following postulates. Postulate I. Space is metric (in the sense of Fréchet). Postulate II. Through any two points a line segment can be drawn, locally uniquely. Postulate III. Any isometry between subsets of space can be extended to a self-isometry of all space. The outline of the proof has been presented elsewhere (G. Birkhoff [2 ]('))• Earlier, H. Busemann [l] had shown that if, further, exactly one straight line connected any two points, and if bounded sets were compact, then the space was Euclidean or hyperbolic; moreover for this, Postulate III need only be assumed for triples of points. Later, Busemann extended the result to elliptic space, assuming that exactly one geodesic connected any two points, that bounded sets were compact, and that Postulate III held locally for triples of points (H. Busemann [2, p. 186]). We recall Lie's celebrated proof that any Riemannian variety having local free mobility is locally Euclidean, spherical, or hyperbolic. Since our Postu- late II, unlike Busemann's straight line postulates, holds in any Riemannian variety in which bounded sets are compact, our result may be regarded as freeing Lie's theorem for the first time from its analytical hypotheses and its local character. What is more important, we use direct geometric arguments, where Lie used sophisticated analytical and algebraic methods(2). Our methods are also far more direct and elementary than those of Busemann, who relies on a deep theorem of Hurewicz.
    [Show full text]
  • Introduction to Algebraic Topology
    Introduction to Algebraic Topology Johnny Nicholson University College London https://www.ucl.ac.uk/~ucahjni/ Lecture 2. Invariants January 26th, 2021 Recap We considered two types of metric spaces: n I Subsets of R I 1 and 2-dimensional cell complexes Definition Two metric spaces X and Y are homotopy equivalent (X ' Y ) if there exists maps f : X ! Y and g : Y ! X such that f ◦ g ' idY and g ◦ f ' idX We say that f : X ! Y is a homotopy equivalence I g ◦ f ' idX means there exists a a one-parameter family of maps Ht : X ! X such that H0 = g ◦ f and H1 = idX and which varies continuously for t 2 [0; 1] We will show that the circle S1 is homotopy equivalent to the punctured complex plane C n f0g 1 iθ We will take S = fe : θ 2 [0; 2π]g ⊆ C Exercise: Prove that this is homeomorphic to 2 2 2 f(x; y): x + y = 1g ⊆ R (our previous definition) 1 1 We need to find f : S ! C n f0g and g : C n f0g ! S such that f ◦ g ' idCnf0g and g ◦ f ' idS1 Fortunately, there are only two sensible choices for f and g: 1 iθ iθ I f : S ! C n f0g; e 7! e 1 iθ iθ I g : C n f0g ! S ; re 7! e (where r 6= 0) We have g ◦ f = idS1 and so need to show that f ◦ g ' idCnf0g Since f g reiθ = eiθ, we need to find a continuous one-parameter family of maps Ht : C n f0g ! C n f0g such that iθ iθ iθ iθ H0 re = e ; H1 re = re iθ t iθ One example is the function Ht re = r e : iθ iθ H1 re = re iθ t iθ Ht e = r e iθ iθ H0 re = e 1 Hence S ' C n f0g How do we prove that two spaces are not homotopy equivalent? 1 For example, is S ' R? One approach is: 1 I Suppose there exists continuous
    [Show full text]
  • Math 131: Introduction to Topology 1
    Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.
    [Show full text]
  • Topological Vector Spaces
    An introduction to some aspects of functional analysis, 3: Topological vector spaces Stephen Semmes Rice University Abstract In these notes, we give an overview of some aspects of topological vector spaces, including the use of nets and filters. Contents 1 Basic notions 3 2 Translations and dilations 4 3 Separation conditions 4 4 Bounded sets 6 5 Norms 7 6 Lp Spaces 8 7 Balanced sets 10 8 The absorbing property 11 9 Seminorms 11 10 An example 13 11 Local convexity 13 12 Metrizability 14 13 Continuous linear functionals 16 14 The Hahn–Banach theorem 17 15 Weak topologies 18 1 16 The weak∗ topology 19 17 Weak∗ compactness 21 18 Uniform boundedness 22 19 Totally bounded sets 23 20 Another example 25 21 Variations 27 22 Continuous functions 28 23 Nets 29 24 Cauchy nets 30 25 Completeness 32 26 Filters 34 27 Cauchy filters 35 28 Compactness 37 29 Ultrafilters 38 30 A technical point 40 31 Dual spaces 42 32 Bounded linear mappings 43 33 Bounded linear functionals 44 34 The dual norm 45 35 The second dual 46 36 Bounded sequences 47 37 Continuous extensions 48 38 Sublinear functions 49 39 Hahn–Banach, revisited 50 40 Convex cones 51 2 References 52 1 Basic notions Let V be a vector space over the real or complex numbers, and suppose that V is also equipped with a topological structure. In order for V to be a topological vector space, we ask that the topological and vector spaces structures on V be compatible with each other, in the sense that the vector space operations be continuous mappings.
    [Show full text]
  • Limits and Topology of Metric Spaces
    LIMITS AND TOPOLOGY OF METRIC SPACES PAUL SCHRIMPF SEPTEMBER 26, 2013 UNIVERSITY OF BRITISH COLUMBIA ECONOMICS 526 This lecture focuses on sequences, limits, and topology. Similar material is covered in chapters 12 and 29 of Simon and Blume, or 1.3 of Carter. 1. SEQUENCES AND LIMITS f g f g¥ f g A sequence is a list of elements, x1, x2, ... or xn n=1 or sometimes just xn . Al- though the notation for a sequence is similar to the notation for a set, they should not be confused. Sequences are different from sets in that the order of elements in a sequence matters, and the same element can appear many times in a sequence. Some examples of 2 R sequences with xi include (1) f1, 1, 2, 3, 5, 8, ...g f 1 1 1 g (2) 1, 2 , 3 , 4 , ... f 1 −2 3 −4 5 g (3) 2 , 3 , 4 , 5 , 6 , ... Some sequences, like 2, have elements that all get closer and closer to some fixed point. We say that these types of sequences converge. A sequence that does not converge di- verges. Some divergent sequences like 1, increase without bound. Other divergent se- quences, like 3, are bounded, but they do not converge to any single point. To analyze sequences with elements that are not necessarily real numbers, we need to be able to say how far apart the entries in the sequence are. Definition 1.1. A metric space is a set, X, and function d : X × X ! R called a metric (or distance) such that 8x, y, z 2 X (1) d(x, y) > 0 unless x = y and then d(x, x) = 0 (2) (symmetry) d(x, y) = d(y, x) (3) (triangle inequality) d(x, y) ≤ d(x, z) + d(z, y).
    [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]