DOCSLIB.ORG

On Uniform Limit Theorem and Completion of Probabilistic Metric Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Uniform Limit Theorem and Completion of Probabilistic Metric Space Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 455 - 461 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4120 On Uniform Limit Theorem and Completion of Probabilistic Metric Space Abderrahim Mbarki National school of Applied Sciences P.O. Box 669, Oujda University, Morocco MATSI Laboratory Abedelmalek Ouahab Department of Mathematics Oujda university, 60000 Oujda Morocco MATSI Laboratory Rachid Naciri MATSI Laboratory Oujda university, 60000 Oujda Morocco Copyright c 2014 Abderrahim Mbarki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited. Abstract A necessary and sufficient condition for a probabilistic metric space to be complete is given and the uniform limit theorem [2] is generalized to probabilistic metric space. Mathematics Subject Classification: 54A40, 54E50, 54D65 Keywords: Uniform Limit Theorem, Completion of PM space 1 Introduction and Preliminaries Our terminology and notation for probabilistic metric spaces conform of that B. Schweizer and A. Sklar [3, 4]. A nonnegative real function f defined on 456 A. Mbarki, A Ouahab and R. Naciri R+ ∪ {∞} is called a distance distribution function (briefly, a d.d.f.) if it is nondecreasing, left continuous on (0, ∞). with f(0) = 0 and f(∞) = 1. The set of all d.d.f’s will be denoted by Δ+; and the set of all f in Δ+ for which + + lims→∞ f(s)=1byD .Fora ∈ [0, ∞), the element a ∈ D is defined as 0ifx ≤ a εa(x)= 1ifx>a and 0, 0 ≤ x<∞, ε∞(x)= 1,x= ∞. By setting f ≤ g whenever f(x) ≤ g(x) for all x ∈ [0, ∞), one introduces a + + natural ordering in , in this ordering the d.d.f 0 is the maximal of . + Convergence in is assumed to be weakly convergence, i.e fn → f if and only if fn(x) → f(x) at each continuity point x of f. Definition 1.1 Let f and g be in Δ+ , let h be in (0, 1], and let (f,g; h) denote the condition 0 ≤ g(x) ≤ f(x + h)+h 1 for all x in (0, h ). + + The modified L´evy distance is the function dL defined on Δ × Δ by dL(f,g) = inf{h : both (f,g; h) and (g, f; h) hold}. + Note that for any f and g in Δ , both (f,g; 1) and (g, f; 1) hold, whence dL is well-defined function and dL(f,g) ≤ 1. Lemma 1.2 [3] For any f in Δ+ dL(f,ε0) = inf{h :(f,ε0; h) holds} = inf{h : lims→h+ f(s) > 1 − h}; and for any t>0, f(t) > 1 − t iff dL(f,ε0) <t. + If f and g are in Δ and f ≤ g, then dL(g, ε0) ≤ dL(f,ε0). A t-norm is a binary operation on [0, 1] which is associative, commutative, nondecreasing in each place and has 1 as identity. Three typical examples of continuous t-norms are: Tp(a, b)=ab , TM (a, b)=Min(a, b) and TL(a, b)=max{a + b − 1, 0}. Uniform limit theorem and completion of PM space 457 A triangle function is a mapping τ : + ×+ →+ that is associative, commutative, nondecreasing in each place and has 0 as identity. Typical continuous triangle function is τT (f,g)(t)=sup{T (f(u),g(v)) : u + v = t}. where T is a continuous t-norm. Definition 1.3 A probabilistic metric space (briefly,PM space) is a triple (X, F, τ) where X is a nonempty set , F is a function from X × X into +, τ is a continuous triangle function , and the following conditions are satisfied for all x, y, z in X, (i) F (x, x)=ε0. (ii) F (x, y) = ε0 if x = y. (iii) F (x, y)=F (y, x). (iv) F (x, z) ≥ τ(F (x, y),F(y, z)). Throughout this paper, we shall frequently denoted F (x, y)byFxy. Definition 1.4 Let (M,F) be a probabilistic semimetric space (i.e. (i), (ii) and (iii) are satisfied). For p in M and t>0, the strong t-neighborhood of p is the set Np(t)={q ∈ M : Fpq(t) > 1 − t}. and the strong neighborhood system for M is {Np(t); p ∈ M, t > 0}. Lemma 1.5 [3] Let (M,F,τ) be a PM space. If τ is continuous, then the family Υ consisting of ∅ and all unions of elements of strong neighborhood system for M determines a Hausdorff topology for M. An immediate consequence of Lemma 1.5 is that the family {Np(t):t>0} is a neighborhood system Definition 1.6 [3] Let {xn} be a sequence in a PM space (X, F, τ). Then (i) The sequence {xn} is said to be convergent to x ∈ X if for all t>0 there exist a positif integer N such that Fxnx(t) > 1 − t for n ≥ N. (ii) The sequence {xn} is called a Cauchy sequence if for all t>0 there exist a positif integer N such that Fxnxm (t) > 1 − t for n, m ≥ N. (iii) APMspace(X, F, τ) is said to be complete if each Cauchy sequence in X is convergent to some point x in X. 458 A. Mbarki, A Ouahab and R. Naciri Lemma 1.7 [3] Let {xn} be a sequence in a PM space (X, F, τ). Then (i) The sequence {xn} to be convergent to x ∈ X iff limn→∞ Fxnx = ε0. (ii) The sequence {xn} is a Cauchy sequence iff limn,m→∞ Fxnxm = ε0. Lemma 1.8 [3] If (X, F, τ) is a PM space, (xn) and (yn) are sequences such that xn → x and yn → y, then Fxnyn → Fxy. Here and in the sequel, when we speak about a probabilistic metric space (M,F,τ), we always assume that τ is continuous and M be endowed with the topology Υ. Recall the Definition of probabilistic diameter of a set in PM space. Definition 1.9 [3] Let A a nonempty subset of a PM space (X, F, τ). The probabilistic diameter of A is the function defined on [0, ∞] by DA(∞)=1 − and DA(t)=L ϕA(t) on [0, ∞). Where ϕA(t) = inf{Fpq(t)|p, q in A} + It is immediate that DA is in Δ for any A ⊂ M. Lemma 1.10 [3] The probabilistic diameter DA has the following proper- ties: i. DA = ε0 iff A is a singleton set. ii. If A ⊂ B, then DA ≥ DB. iii. For any p, q ∈ A, Fpq ≥ DA. iv. If A = {p, q}, then DA = Fpq. v. If A ∩ B is nonempty, then DA∪B ≥ τ(DA,DB). vi. DA = DA, where A is the strong closure of A. The diameter of a nonempty set A in a metric space is either finite or infinite; accordingly, A is either bounded or unbounded. In a PM space, on the other hand , there are three distinct possibilities. These are captured in Definition 1.11 [3] A nonempty set A in a PM space is + (i) Bounded if DA is in D . (ii) Semi-bounded if 0 < limt→∞ DA(t) < 1. (iii) Unbounded if limt→∞ DA(t)=0. Example 1.12 Let (M,d) be a metric space. Define F d : M × M → Δ+ the probalistic metric induced by d as d Fpq = εd(p,q). Uniform limit theorem and completion of PM space 459 d It is easy to check that (M,F ,τMin) is a PM (Menger) space, and Np(t)={q ∈ M : d(p, q) <t}, for t in (0, 1). So (M,F,τMin) is a complete PM space if and only if (M,d) is a complete metric space. Moreover, for A a nonempty subset of M we have DA = εdiam(A), where diam(A)=sup{d(p, q): p, q ∈ A}. Let us now state our results. 2 Completion of probabilistic metric space Theorem 2.1 APMspace(M,F,τ) is complete if and only if for each creasing sequence of nonempty closed sets {Fn} such that DFn −→ 0 have nonempty intersection. Proof. Let {xn} be a Cauchy sequence in M. Consider An = {xi : i ≥ n}.It { } is obvious that An that is a creasing sequence. Now we claim that DAn ε0. For given s>0. Let s>t>0, since {xn} is Cauchy sequence then ∀>0 there exists N ∈ IN such that for all n, p ≥ NFxnxp (t) ≥ 1 − . So ϕAn (t) ≥ 1 − . for any n ≥ N. It follows that ϕAn (t ) ≥ 1 − . for any s>t >t>0. Letting t → s, we obtain DAn (s) ≥ 1 − . It follows from Lemma 1. 10 that → DAn (s)=DAn (s) 1. Since s is arbitrarily positive number. This clearly means that D ε0. An ∅ ∈ ≥ Hence by hypothesis n An = . Take x n An then Fxnx DAn .So Fxnx ε0, this means that xn → x as n →∞. Hence (M,F,τ) is complete PM space Conversely, suppose that (M,F,τ) is complete PM space and {Fn} is a creasing sequence of nonempty closed sets of M such that DFn 0. Since Fn = ∅ 460 A. Mbarki, A Ouahab and R. Naciri there exists xn ∈ Fn. Continuing in this manner we can construct by induction a sequence {xn} such that for each n ∈ IN ,xn ∈ Fn. Next we claim that {xn} is a Cauchy sequence. Indeed, lets n>p>0, then Fxnxp ≥ DFp . which implies that Fxnxp ε0 as n, p →∞, this means that {xn} is a Cauchy sequence, since (M,F,τ) is complete PM space then there exists x ∈ M such that xn → x.
Load more

Recommended publications
  • Elementary Real Analysis
    Elementary Real Analysis MAT 2125 Winter 2017 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Contents Preface iii 1 The real numbers1 1.1 Algebraic structure: the field axioms......................1 1.2 Order structure..................................3 1.3 Bounds and the completeness axiom......................4 1.4 Natural numbers and induction.........................7 1.5 Rational numbers................................. 10 1.6 Absolute value: the metric structure on R ................... 11 1.7 Constructing the real numbers.......................... 12 2 Sequences 13 2.1 Limits....................................... 13 2.2 Properties of limits................................ 17 2.3 Monotonic convergence criterion......................... 20 2.4 Subsequences................................... 22 2.5 Cauchy sequences................................. 25 2.6 Limit inferior and limit superior......................... 27 3 Series 31 3.1 Definition and basic properties.......................... 31 3.2 Convergence tests................................. 35 3.3 Absolute convergence............................... 40 4 Topology of Rd 42 4.1 Norms....................................... 42 4.2 Convergence in Rd ................................ 45 4.3 Open and closed sets............................... 48 4.4 Compact sets................................... 54 5 Continuity 59 5.1 Limits......................................
    [Show full text]
  • Complex Analysis
    University of Cambridge Mathematics Tripos Part IB Complex Analysis Lent, 2018 Lectures by G. P. Paternain Notes by Qiangru Kuang Contents Contents 1 Basic notions 2 1.1 Complex differentiation ....................... 2 1.2 Power series .............................. 4 1.3 Conformal maps ........................... 8 2 Complex Integration I 10 2.1 Integration along curves ....................... 10 2.2 Cauchy’s Theorem, weak version .................. 15 2.3 Cauchy Integral Formula, weak version ............... 17 2.4 Application of Cauchy Integration Formula ............ 18 2.5 Uniform limits of holomorphic functions .............. 20 2.6 Zeros of holomorphic functions ................... 22 2.7 Analytic continuation ........................ 22 3 Complex Integration II 24 3.1 Winding number ........................... 24 3.2 General form of Cauchy’s theorem ................. 26 4 Laurent expansion, Singularities and the Residue theorem 29 4.1 Laurent expansion .......................... 29 4.2 Isolated singularities ......................... 30 4.3 Application and techniques of integration ............. 34 5 The Argument principle, Local degree, Open mapping theorem & Rouché’s theorem 38 Index 41 1 1 Basic notions 1 Basic notions Some preliminary notations/definitions: Notation. • 퐷(푎, 푟) is the open disc of radius 푟 > 0 and centred at 푎 ∈ C. • 푈 ⊆ C is open if for any 푎 ∈ 푈, there exists 휀 > 0 such that 퐷(푎, 휀) ⊆ 푈. • A curve is a continuous map from a closed interval 휑 ∶ [푎, 푏] → C. It is continuously differentiable, i.e. 퐶1, if 휑′ exists and is continuous on [푎, 푏]. • An open set 푈 ⊆ C is path-connected if for every 푧, 푤 ∈ 푈 there exists a curve 휑 ∶ [0, 1] → 푈 with endpoints 푧, 푤.
    [Show full text]
  • Proofs-Munkres-35.Pdf
    Introduction to Topology Chapter 4. Countability and Separation Axioms Section 35. The Tietze Extension Theorem—Proofs of Theorems September 10, 2016 () Introduction to Topology September 10, 2016 1 / 10 Table of contents 1 Theorem 35.1. The Tietze Extension Theorem () Introduction to Topology September 10, 2016 2 / 10 Step 1. First, we consider the case f : A → [−r, r]. We construct continuous g : X → R such that |g(x)| < r/3 for all x ∈ X and |g(a) − f (a)| < 2r/3 for all a ∈ A. For the construction, define −1 I1 = [−r, −r/3], I2 = [−r/3, r/3], and I3 = [r/3, r]. Let B = f (I1) and −1 C = f (I3) (subsets of A. Since f is continuous then B and C are closed (by Theorem 18.1(3)) and are disjoint. Theorem 35.1. The Tietze Extension Theorem Theorem 35.1. The Tietze Extension Theorem Theorem 35.1. The Tietze Extension Theorem. Let X be a normal space. Let A be a closed subspace of X . (a) Any continuous function of A into the closed interval [a, b] ⊂ R may be extended to a continuous function on all of X into [a, b]. (b) Any continuous function of A into R may be extended to a continuous function on all of X into R. Proof. We follow Munkres’ three step proof and construct the desired function as a limit of a sequence of functions. () Introduction to Topology September 10, 2016 3 / 10 For the construction, define −1 I1 = [−r, −r/3], I2 = [−r/3, r/3], and I3 = [r/3, r].
    [Show full text]
  • On Some Results of Analysis in Metric Spaces and Fuzzy
    ON SOME RESULTS OF ANALYSIS IN METRIC SPACES AND FUZZY METRIC SPACES by MAGGIE APHANE submitted in fulfillment of the requirements for the degree of MASTER OF SCIENCE in the subject MATHEMATICS at the UNIVERSITY OF SOUTH AFRICA Supervisor: PROF SP MOSHOKOA DECEMBER 2009. Contents Table of Contents . ii Acknowledgements . iii Summary . iv Symbols . 1 1 Introduction. 2 1.1 Convergence and completeness in metric spaces. 2 1.2 Continuity and uniform continuity in metric spaces. 10 2 Fundamental properties of fuzzy metric spaces. 16 2.1 Basic notions on fuzzy metric spaces. 17 2.2 Topology and fuzzy metric spaces. 28 3 Further properties on fuzzy metric spaces. 39 3.1 Complete fuzzy metric spaces. 39 3.2 Separability and uniform convergence in fuzzy metric spaces. 45 4 Some properties on fuzzy pseudo metric spaces. 49 4.1 Fuzzy pseudo metric spaces and some properties. 49 4.2 Fuzzy pseudo metric spaces and uniformities. 56 i 4.3 Fuzzy metric identification. 59 4.4 Uniformly continuous maps and extension of t−nonexpansive maps. 65 ii Acknowledgements I would like to express my sincere gratitude and deep appreciation to the following: • My Creator who was always available when I needed Him. • My supervisor Prof S.P. Moshokoa, for his positive attitude, comments, helpful suggestions and expert guidance throughout my studies until the completion of the dissertation. • Prof T. Dube for revising my work. • I would also like to thank J-P. Motsei, E.T. Motlotle, A.S. Kubeka and the late Dr C.M. Kumile, for their generous support and assistance.
    [Show full text]
  • ON FUZZY UNIFORM CONVERGENCE Jae Eung Kong
    Kangweon-Kyungki Math. Jour. 5 (1997), No. 1, pp. 1–8 ON FUZZY UNIFORM CONVERGENCE Jae Eung Kong and Sung Ki Cho Abstract. In this note, we study on fuzzy uniform convergences of sequences of fuzzy numbers, and sequences of fuzzy functions. 1. Introduction Zhang [1] provided “The Cauchy criterion for sequences of fuzzy numbers” under a restricted condition. In this note, we prove the criterion of fuzzy uniform convergences for fuzzy numbers and fuzzy functions. 2. Preliminaries All fuzzy sets, considered in this paper, are functions defined in the set R of real numbers to [0, 1]. Definition 2.1. [1] A fuzzy set A is called a fuzzy number if the following conditions are satisfied: (1) there exists x ∈ R such that A(x) = 1; (2) for any λ ∈ (0, 1], the set {x|A(x) ≥ λ} is a closed interval, − + denoted by [Aλ ,Aλ ]. Note that every fuzzy point a1 (a ∈ R) defined by 1 for x = a a1(x) = 0 for x 6= a Received July 15, 1996. 1991 Mathematics Subject Classification: 40A30. Key words and phrases: fuzzy Cauchy sequence, uniform fuzzy Cauchy se- quence, pointwise fuzzy convergent sequence, uniform fuzzy convergent sequence, uniform fuzzy convergent series. 2 Jae Eung Kong and Sung Ki Cho is a fuzzy number. Let F(R) be the set of all fuzzy numbers. Remark that for any A ∈ F(R), A = sup λχ[A−,A+], λ∈[0,1] λ λ where each χ − + denotes the characteristic function. For notational [Aλ ,Aλ ] − + convenience, we shall denote χ − + by [A ,A ]. [Aλ ,Aλ ] λ λ Definition 2.2.
    [Show full text]
  • Continuity in Vector Metric Spaces
    CONTINUITY IN VECTOR METRIC SPACES CUNEYT¨ C¸EVIK˙ Abstract. We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems. 1. Introduction and Preliminaries Let E be a Riesz space. If every nonempty bounded above (countable) subset of E has a supremum, then E is called Dedekind (σ-)complete. A subset in E is called order bounded if it is bounded both above and below. We write an ↓ a if (an) is a decreasing sequence in E such that inf an = a. The Riesz space E is −1 said to be Archimedean if n a ↓ 0 holds for every a ∈ E+. A sequence (bn) is said to be o-convergent (or order convergent) to b if there is a sequence (an) in E such that an ↓ 0 and |bn − b| ≤ an for all n, where |a| = a ∨ (−a) for any a ∈ E. o We will denote this order convergence by bn −→ b. Furthermore, a sequence (bn) is said to be o-Cauchy if there exists a sequence (an) in E such that an ↓ 0 and |bn − bn+p|≤ an for all n and p. The Riesz space E is said to be o-Cauchy complete if every o-Cauchy sequence is o-convergent. An operator T : E → F between two Riesz spaces is positive if T (x) ≥ 0 for all x ≥ 0. The operator T is said to be order bounded if it maps order bounded subsets of E to order bounded subsets of F .
    [Show full text]
  • MTH 303: Real Analysis
    MTH 303: Real Analysis Semester 1, 2013-14 1. Real numbers 1.1. Decimal and ternary expansion of real numbers. (i) [0; 1] and R are uncountable. (ii) Cantor set is an uncounbtable set of measure zero. 1.2. Order properties of real numbers. (i) Least upper bound axiom. (ii) Every subset of R that has a lower bound also has a greatest lower bound. (iii) Archimedean property. (iv) Density of Q. 2. Sequences of real numbers 2.1. Limits of sequences. (i) Sequences and their limits with examples. (ii) A sequence in R has at most one limit. (iii) A sequence converges if and only if its tail converges. (iv) Let (xn) be a sequence of real numbers and let x 2 R. If (an) is a sequence of positive reals with lim(an) = 0 and if we have a constant C > 0 and some m 2 N such that jxn − xj ≤ Can, for n ≥ m, then lim(xn) = x. 2.2. Limit Therorems. (i) Bounded sequences with examples. (ii) A convergent sequence of real numbers is bounded. (iii) Convergence of the sum, difference, product, and quotient of sequences. (iv) The limit of a non-negative sequence is non-negative. (v) Squeeze Theorem: Suppose that X = (xn), Y = (yn), and Z = (zn) are sequences such that xn ≤ yn ≤ zn, 8 n, and lim(xn) = lim(zn). Then Y converges and lim(xn) = lim(yn) = lim(zn). (vi) Let X = (x ) be a sequence of positive reals such that L = lim xn+1 . n xn If L < 1, then X converges and lim(xn) = 0.
    [Show full text]
  • 5 Urysohn's Lemma and Applications
    FTAR – 1o S 2014/15 Contents 1 Topological spaces 2 1.1 Metric spaces . .2 1.2 Topologies and basis . .2 1.3 Continuous functions . .2 1.4 Subspace topologies . .2 1.5 Product topologies . .2 1.6 Quotient spaces . .3 2 Connected spaces 5 3 Countability axioms 5 4 Separation axioms 8 5 Urysohn’s Lemma and applications 12 5.1 Urysohn’s Lemma . 12 5.2 Completely regular spaces . 14 5.3 Urysohn’s Metrization theorem . 15 5.4 Tietze extension theorem . 16 6 Compactness 18 6.1 Compact spaces . 18 6.2 Products . 21 6.3 Locally compact spaces . 23 6.4 Compactness in metric spaces . 23 6.5 Limit point and sequential compactness . 23 7 Complete metric spaces and function spaces 26 7.1 Completeness . 26 1 Notas de Apoio as` Aulas Teoricas´ de FTAR Prof. Responsavel:´ Catarina Carvalho 1o Semestre de 2015/2016 1 Topological spaces 1.1 Metric spaces Neighborhoods Continuous functions Convergence sequences Cauchy sequences 1.2 Topologies and basis 1.3 Continuous functions 1.4 Subspace topologies rel metric 1.5 Product topologies rel metric 2 FTAR – 1o S 2014/15 1.6 Quotient spaces Definition 1.1. We say that Y is a quotient of X (with respect to p) if p : X Y is surjective 1 ! and the topology on Y is such that U Y open, if, and only if p− (U) is open in X. This topology is called a quotient topology on⊂ Y (induced by p), and p is called a quotient map. 1 Note that since p is surjective, we always have p(p− (B)) = B, for all B Y.
    [Show full text]
  • Real Analysis Qual Prep Week 1: Preliminaries
    Real Analysis Qual Prep Week 1: Preliminaries D. Zack Garza 1 Contents Table of Contents Contents Table of Contents 2 1 Week 1: Preliminaries3 1.1 Topics............................................3 1.2 Background / Warmup / Review.............................3 1.2.1 Metric Spaces / Topology.............................4 1.2.2 Sequences......................................4 1.2.3 Series........................................5 1.2.4 Continuity and Discontinuity...........................5 1.3 Exercises..........................................5 1.4 Qual Questions.......................................6 Table of Contents 2 1 Week 1: Preliminaries 1 Week 1: Preliminaries E 1.1 Topics e • Concepts from Calculus – Mean value theorem – Taylor expansion – Taylor’s remainder theorem – Intermediate value theorem – Extreme value theorem – Rolle’s theorem – Riemann integrability • Continuity and uniform continuity – Pathological functions and sequences of functions • Convergence – The Cauchy criterion – Uniform convergence – The M-Test • Fσ and Gδ sets, • Nowhere density, • Baire category theorem, • Heine-Borel • Normed spaces • Series and sequences, – Convergence – Small tails, – limsup and liminf, – Cauchy criteria for sums and integrals • Basic inequalities (triangle, Cauchy-Schwarz) • Weierstrass approximation • Variation and bounded variation E 1.2 Background / Warmup / Review e • Derive the reverse triangle inequality from the triangle inequality. • Let E ⊆ R. Define sup E and inf E. • What is the Archimedean property? Week 1: Preliminaries
    [Show full text]
  • Baire Categories, Continuous Functions, and Approximations
    BAIRE CATEGORIES, CONTINUOUS FUNCTIONS, AND APPROXIMATIONS ANDREW CHEN Abstract. This paper is an exploration into continuous functions and how they can be approximated. We first gain insight into the nature of continuous functions and demonstrate why most continuous functions are nowhere differentiable using the Baire Category Theorem. We then discuss two methods of approximating continuous functions: the first of which is using polynomials via the Weierstrass Approximation Theorem, and the second of which is using neural networks via the Universal Approximation Theorem. Contents 1. Introduction 1 2. Continuous Functions and Differentiability 1 2.1. Introducing the Baire Category Theorem 2 2.2. Applying the Baire Category Theorem to the Set of Continuous Functions 4 3. Weierstrass Approximation Theorem 6 4. The Universal Approximation Theorem 9 Acknowledgments 14 References 15 1. Introduction For many years, it was widely thought by mathematicians that most (if not all) continuous functions are differentiable, due to intuitive notions of continuity and differentiability. While calculus was invented in the late 1600s, the first known example of a non-differentiable continuous function was not discovered until around year 1830, by Bernard Bolzano, and the first published example was not presented until 1872, by Karl Weierstrass [9]. Due to later advances in the field of topology, we now know that most continuous functions are in fact nowhere differentiable. Section 2 will explore a proof of this, which utilizes an important theorem in topology called the Baire Category Theorem. We will then explore methods of approximating continuous functions, even those that are nowhere differentiable. Section 3 will explore a probabilistic proof of the Weierstrass Approximation Theorem, which will show that all continuous functions f : [0; 1] ! R can be approximated arbitrarily well using polynomials.
    [Show full text]
  • Lecture Notes on Topology for MAT3500/4500 Following J. R. Munkres’ Textbook
    Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres' textbook John Rognes November 21st 2018 Contents Introduction v 1 Set Theory and Logic 1 1.1 (x1) Fundamental Concepts . 1 1.1.1 Membership . 1 1.1.2 Inclusion and equality . 2 1.1.3 Intersection and union . 2 1.1.4 Difference and complement . 2 1.1.5 Collections of sets, the power set . 3 1.1.6 Arbitrary intersections and unions . 3 1.1.7 Cartesian products . 4 1.2 (x2) Functions . 4 1.2.1 Domain, range and graph . 4 1.2.2 Image, restriction, corestriction . 5 1.2.3 Injective, surjective, bijective . 5 1.2.4 Composition . 6 1.2.5 Images of subsets . 6 1.2.6 Preimages of subsets . 7 1.3 (x5) Cartesian Products . 8 1.3.1 Indexed families . 8 1.3.2 General intersections and unions . 8 1.3.3 Finite cartesian products . 9 1.3.4 Countable cartesian products . 9 1.3.5 General cartesian products . 10 1.4 (x6) Finite Sets . 10 1.4.1 Cardinality . 10 1.4.2 Subsets . 11 1.4.3 Injections and surjections . 12 2 Topological Spaces and Continuous Functions 13 2.1 (x12) Topological Spaces . 13 2.1.1 Open sets . 13 2.1.2 Discrete and trivial topologies . 13 2.1.3 Finite topological spaces . 14 2.1.4 The cofinite topology . 15 2.1.5 Coarser and finer topologies . 16 2.1.6 Metric spaces . 17 2.2 (x13) Basis for a Topology . 18 2.2.1 Bases . 18 i 2.2.2 Comparing topologies using bases .
    [Show full text]