Prof. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.
|
|
- Dorcas Rose
- 7 years ago
- Views:
Transcription
1 Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then {f(x n)} n=1 is a convergent sequence in Y Show that a convergent sequence {x n } n=1 in X is Cauchy Give an example of a metric space (X, d) and a Cauchy sequence {x n } n=1 does not converge. (Enough to quote previous homework problem.) from X that Give an example of a continuous f : X Y that does not preserve Cauchy sequences Show that if f is uniformly continuous, then f preserves Cauchy sequences Prove or give a counterexample. If f preserves Cauchy sequences, then f is uniformly continuous Show the following. f is uniformly continuous if {x n } n=1 and {z n} n=1 are sequences from X s.t. d(x n, z n ) 0, then ρ(f(x n ), f(z n )) 0. Remark 0.1. Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. (a) Recall [K, Prop. 2.25]. f is continuous at x X if {x n } n=1 X and x n x, then f(x n ) f(x). (b) Definition. f preserves convergent sequences if {x n } n=1 X converges, say x n x, then f(x n ) f(x). (c) So: 1 f is continuous on X f preserves convergent sequences. (d) Definition. f preserves Cauchy sequences if {x n } n=1 X is Cauchy, then {f(x n)} n=1 Y is Cauchy. Remark 0.2. For a dicussion of this problem, visit the below website. Proof. LTGBG This is true. Here comes the proof Let f preserve convergent sequences. Consider a sequence {x n } n=1 from X that converges to some x in X. Then {f(x n )} n=1 converges in Y, specifically to f(x) Assume that if {x n } n=1 is a convergent sequence in X then {f(x n)} n=1 is a convergent sequence in Y. Let {x n } n=1 X be converge sequence, say x n x. We WTS that f(x n ) f(x). 1 Follows directly from (a) and (b). Page 1 of 6
2 Define { x n } n=1 by x 2n 1 x 2n = x n = x for each n N. Then x n x. So by assumption, there is y Y s.t. f( x n ) y. By considering the subsequence {f( x 2n )} n=1 of {f( x n)} n=1 we get that y = lim f( x 2n) = n lim f(x) = f(x). n Thus f(x n ) f(x) Let {x n } n=1 be a convergent sequence in X, say x n x. Fix ɛ > 0. Find N ɛ N s.t. if n N ɛ then d(x n, x) < ɛ 2. Let n, m > N ɛ. Then Thus {x n } n=1 is a Cauchy sequence in X Recall Homework # 8. d(x n, x m ) d(x n, x) + d(x, x m ) < ɛ 2 + ɛ 2 = ɛ. Trench, 8.1, # 25, p Consider the set C[a, b] of continuous real functions on the interval [a, b]. (a) Show that is a norm on C[a, b]. f = b a f(x) dx (b) Show that the sequence {f n } defined by is a Cauchy sequence in (C[a, b], ). f n (x) = (c) Show that (C[a, b], ) is not complete. ( ) x a n b a So we can take the metric space (X, d) to be (C[a, b], d) where d(f, g) := f g and take {f n } n=1 as in (b) above Let (X, d) := ((0, 1), ) and (Y, ρ) := (R, ). Let f(x) := 1/x. Clearly, f : X Y is a continuous function. Consider the sequence {x n } n=1 in X where x n := 1/n. Then {x n } n=1 is Cauchy in X since 0 d(x n, x m ) = 1 n 1 m 1 n + 1 m n,m 0. But {f(x n )} n=1 = {n} n=1 is not Cauchy in Y since ρ(f(x n), f(x n+1 )) = 1 for each n N Let f be uniformly continuous. Consider a Cauchy sequence {x n } n=1 in X. We WTS that {f(x n )} n=1 is a Cauchy sequence in Y. Page 2 of 6
3 Fix ɛ > 0. Since f is uniformly continuous, there exists δ ɛ > 0 such that if x, x X and d(x, x) < δ ɛ, then ρ(f(x), f( x)) < ɛ. (1) Furthermore, since {x n } n is Cauchy, there exists N ɛ N such that if n, m N ɛ, then d(x n, x m ) < δ ɛ. (2) Now let n, m N ɛ. It follows from (1) and (2) that ρ(f(x n ), f(x m )) < ɛ. Therefore, {f(x n )} n is Cauchy in Y Preservation of Cauchy sequences does not imply uniform continuity. Let (X, d) = (Y, ρ) = (R, ). Consider the function f : X Y given by f(x) := x 2. Clearly f is continuous on X. To see that f preserves Cauchy sequences, first recall that a sequence in (R, ) is Cauchy if and only if it converges. Next fix a Cauchy sequence {x n } n in X. Then {x n } n is a convergent sequence in X. By Remark 0.1(c) (or HMWK 13.1), since f is continuous on X, the sequence {f(x n )} n converges in Y. Thus {f(x n )} n is a Cauchy sequence in Y. But To see that f is not uniformly continuous on R, fix δ > 0. Let x = 1 δ and x = 1 δ + δ 17. Then x x = δ 17 < δ. f( x) f(x) = ( x) 2 x 2 = δ > So f cannot be uniformly continuous on R Let f is uniformly continuous. Fix sequences {x n } n=1 and {z n} n=1 from X with the property that d(x n, z n ) 0. We WTS that ρ(f(x n ), f(z n )) 0. Fix ɛ > 0. Since f is uniformly continuous on X, there exists δ ɛ > 0 such that if x, x X and d(x, x) < δ ɛ, then ρ(f(x), f( x)) < ɛ. (3) Since d(x n, z n ) 0, there e exists N δɛ N such that if n N δɛ, then d(x n, z n ) < δ ɛ. (4) Now let n N δɛ. It follows from (3) and (4) that if n > N δɛ then ρ(f(x n ), f(z n )) < ɛ. Thus ρ(f(x n ), f(z n )) Suppose that f is not uniformly continuous. 2 It suffices to show there exist sequences {x n } n=1 and {z n} n=1 in X such that d(x n, z n ) 0 but ρ(f(x n ), f(z n )) 0. The negation of the definition of uniform continuity gives that there exists some ɛ 0 > 0 s.t. for each δ > 0, there exist points x δ, z δ in X s.t. d(x δ, z δ ) < δ but ρ(f(x δ ), f(z δ )) ɛ 0. 2 Do you see why we want to argue by contrapositive? Indeed, we are trying to say P Q and Q deals only with sequences but P is continuous. (5) Page 3 of 6
4 Take this ɛ 0 and consider the sequence {δ n } n=1 where δ n := 1/n. Using (5), find x n, z n X s.t. d(x n, z n ) < 1 n but ρ(f(x n ), f(z n )) ɛ 0. Then lim n d(x n, z n ) = 0. But the sequence {ρ(f(x n ), f(z n ))} n=1 ρ(f(x n ), f(z n )) ɛ 0 for each n N. cannot converge to zero since Homework 14 Let (X, d) be a metric space. Let {x n } n=1 be a Cauchy sequence in X that has a convergent subsequence {x n k } k=1, say {x nk } k=1 converges to x X. Show that {x n} n=1 converges to x. Remark 0.3. Loosely speaking: show that a Cauchy sequence in a metric space that has a convergent subsequence converges (of course, to the element that the convergent subsequence converges to). Proof. LTGBG. Fix ɛ > 0. Since {x n } n=1 is Cauchy, there exists some N ɛ N such that if n, m N ɛ, then d(x n, x m ) < ɛ 2. (6) Since {x nk } k=1 converges to x, there exists some K ɛ N such that if k K ɛ, then d(x nk, x) < ɛ 2. (7) Pick k ɛ such that k ɛ K ɛ and n kɛ N ɛ. (8) Fix n N ɛ. It follows from (6), (7), and (8) that d(x n, x) d(x n, x nkɛ ) + d(x nkɛ, x) < ɛ 2 + ɛ 2 = ɛ. Thus {x n } n=1 converges to x. Homework 15 Let (X, d) be a metric space and A X. Consider the metric space (A, d A ) where d A := d A A Show that if A is complete, then A is closed in X Show that if A is totally bounded, then A is bounded. Page 4 of 6
5 Notation 0.4. Throughout this problem, for x 0 X and a 0 A and ɛ > 0, let N X ɛ (x 0 ) := {x X : d(x, x 0 ) < ɛ} N A ɛ (a 0 ) := {a A: d(a, a 0 ) < ɛ}. Remark 0.5. Let s recall definitions. Let (X, d) and (A, d A ) be as in the statement of Homework A is complete provided each Cauchy sequence in A converges to some element in A. 2. A is totally bounded ɛ > 0 A can be covered by finitely many ɛ-balls i.e. ɛ > 0 n N and a 1,... a n A s.t. A = n i=1 N ɛ A (a i ) i.e. ɛ > 0 n N and a 1,... a n A s.t. A n i=1 N ɛ X (a i ). 3. A is bounded provided there exists a 0 A and R > 0 s.t. A N A R (a 0). Proof. LTGBG Let A be complete. We WTS that A is closed in X. Fix a sequence {a n } n=1 that converges to some point in X, say lim n a n = x X. Then {a n } n=1 is a Cauchy sequence in X by Homework 13.2 and thus {a n } n=1 is a Cauchy sequence in A. Since A is complete, by def. of complete, {a n} n=1 converges to some element in a A. Thus x = a A. So by [K, Cor. 2.23], A is closed in X Let A be totally bounded. We WTS that A is bounded. Let ɛ = 17. Since A is totally bounded, by definition of totally bounded, there exists n N and a 1,..., a n A such that Pick an a 0 A. Let A = n i=1n A ɛ (a i ). (9) M := max d(a 0, a i ) 1 i n R := M + ɛ. Clearly 0 < R <. It is enough to show that A N A R (a 0 ). (10) Towards this, fix an a A. By (9), there exists i {1,... n} such that a N A ɛ (a i ). Thus Thus a N A R (a 0). So (10) holds. d(a, a 0 ) d(a, a i ) + d(a i, a 0 ) < ɛ + M = R. Page 5 of 6
6 Homework 16 Let U be an open subset of R. Show U can be expressed as a countable union of disjoint open intervals. Proof. Consider an arbitrary x U. Since U is open, there is an ɛ > 0 so that (x ɛ, x + ɛ) U. Thus the sets {a (, x): (a, x] U} and {b (x, ): [x, b) U} each are nonempty and, by the completeness of R, a x := inf{a (, x): (a, x) U} [, x) b x := sup {b (x, ): [x, b) U} (x, ]. Note the following hold. I x := (a x, b x ) is an open (perhaps unbounded) interval. x I x U. I x is the maximal open interval in U containing x. I.e., if J is any open interval in U containing x, then J I x. Thus U equals the union of open intervals, specifically, U = I x. x U Next we shall show that this union can be expressed as a countable union of disjoint open intervals. Claim 1. For any two points x and y in U, I x and I y are either equal or disjoint. Indeed, if I x I y is nonempty, then I x I y is an open interval in U containing both x and y. But I x and I y are the maximal such open intervals containing x and y, respectively. So it must be that I x = I x I y = I y. So Claim 1 holds. Claim 2. The set (so no repeats allowed) I := {I x : x U} is countable. Indeed, for each I I, we may choose some rational number q Q that lies in I. Since the interval in I are distinct by Claim 1, the rational numbers we choose are distinct from one another. As a result, the set of rational numbers we collected is countable. So Claim 2 holds. Thus U = I I I and this union is a countable union of disjoint open intervals. Page 6 of 6
1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationI. Pointwise convergence
MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationMath 104: Introduction to Analysis
Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More information(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9
Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More informationDISINTEGRATION OF MEASURES
DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationLecture Notes on Measure Theory and Functional Analysis
Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationHow To Find Out How To Calculate A Premeasure On A Set Of Two-Dimensional Algebra
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More information1 Calculus of Several Variables
1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 300-31. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationPolarization codes and the rate of polarization
Polarization codes and the rate of polarization Erdal Arıkan, Emre Telatar Bilkent U., EPFL Sept 10, 2008 Channel Polarization Given a binary input DMC W, i.i.d. uniformly distributed inputs (X 1,...,
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationMetric Spaces. Lecture Notes and Exercises, Fall 2015. M.van den Berg
Metric Spaces Lecture Notes and Exercises, Fall 2015 M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK mamvdb@bristol.ac.uk 1 Definition of a metric space. Let X be a set,
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationTHE CONTRACTION MAPPING THEOREM
THE CONTRACTION MAPPING THEOREM KEITH CONRAD 1. Introduction Let f : X X be a mapping from a set X to itself. We call a point x X a fixed point of f if f(x) = x. For example, if [a, b] is a closed interval
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationINTRODUCTION TO TOPOLOGY
INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology
More information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationLECTURE NOTES IN MEASURE THEORY. Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12)
1 LECTURE NOTES IN MEASURE THEORY Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12) 2 PREFACE These are lecture notes on integration theory for a eight-week
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationDEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More information1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationThe Ergodic Theorem and randomness
The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction
More informationDifferential Operators and their Adjoint Operators
Differential Operators and their Adjoint Operators Differential Operators inear functions from E n to E m may be described, once bases have been selected in both spaces ordinarily one uses the standard
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationUniversity of Miskolc
University of Miskolc The Faculty of Mechanical Engineering and Information Science The role of the maximum operator in the theory of measurability and some applications PhD Thesis by Nutefe Kwami Agbeko
More informationMEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich
MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites
More informationDifferentiating under an integral sign
CALIFORNIA INSTITUTE OF TECHNOLOGY Ma 2b KC Border Introduction to Probability and Statistics February 213 Differentiating under an integral sign In the derivation of Maximum Likelihood Estimators, or
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationLemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.
Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More information