Introduction to Real Analysis, Volume II

Introduction to Real Analysis, Volume II

Basic Analysis II Introduction to Real Analysis, Volume II by Jiríˇ Lebl June 10, 2020 (version 2.3) 2 Typeset in LATEX. Copyright ©2012–2020 Jiríˇ Lebl This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Derivative works must be prominently marked as such. During the writing of these notes, the author was in part supported by NSF grant DMS-1362337. The date is the main identifier of version. The major version / edition number is raised only if there have been substantial changes. Each volume has its own version number. See https://www.jirka.org/ra/ for more information (including contact information, possible updates and errata). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/ra Contents Introduction 5 8 Several Variables and Partial Derivatives 7 8.1 Vector spaces, linear mappings, and convexity ....................7 8.2 Analysis with vector spaces .............................. 19 8.3 The derivative ..................................... 32 8.4 Continuity and the derivative ............................. 42 8.5 Inverse and implicit function theorems ........................ 47 8.6 Higher order derivatives ................................ 56 9 One-dimensional Integrals in Several Variables 61 9.1 Differentiation under the integral ........................... 61 9.2 Path integrals ..................................... 66 9.3 Path independence .................................. 77 10 Multivariable Integral 85 10.1 Riemann integral over rectangles ........................... 85 10.2 Iterated integrals and Fubini theorem ......................... 96 10.3 Outer measure and null sets .............................. 101 10.4 The set of Riemann integrable functions ....................... 109 10.5 Jordan measurable sets ................................ 113 10.6 Green’s theorem .................................... 117 10.7 Change of variables .................................. 121 11 Functions as Limits 125 11.1 Complex numbers ................................... 125 11.2 Swapping limits .................................... 130 11.3 Power series and analytic functions .......................... 138 11.4 The complex exponential and the trigonometric functions .............. 147 11.5 Fundamental theorem of algebra ........................... 153 11.6 Equicontinuity and the Arzelà–Ascoli theorem .................... 156 11.7 The Stone–Weierstrass theorem ............................ 162 11.8 Fourier series ..................................... 174 Further Reading 187 4 CONTENTS Index 189 List of Notation 193 Introduction About this book This second volume of “Basic Analysis” is meant to be a seamless continuation. The chapters are numbered to start where the first volume left off. The book started with my notes for a second- semester undergraduate analysis at University of Wisconsin—Madison in 2012, where I used my notes together with Rudin’s book. The choice of some of the material and some of the proofs are very similar to Rudin, though I do try to provide more detail and context. In 2016, I taught a second-semester undergraduate analysis at Oklahoma State University and heavily modified and cleaned up the notes, this time using them as the main text. In 2018, I taught this course again, this time adding chapter 11 (which I originally wrote for the Wisconsin course). I plan to eventually add some more topics. I will try to preserve the current numbering in subsequent editions as always. The new topics I have planned would add chapters onto the end of the book, or add sections to end of existing chapters, and I will try as hard as possible to leave exercise numbers unchanged. For the most part, this second volume depends on the non-optional parts of volume I, while some of the optional parts are also used. Higher order derivatives (but not Taylor’s theorem itself) are used in 8.6, 9.3, 10.6 . Exponentials, logarithms, and improper integrals are used in a few examples and exercises, and they are heavily used in chapter 11. My own plan for a two-semester course is that some bits of the first volume, such as metric spaces, are covered in the second semester, while some of the optional topics of volume I are covered in the first semester. Leaving metric spaces for the second semester makes more sense as then the second semester is the “multivariable” part of the course. Another possibility for a faster course is to leave out some of the optional parts, go quicker in the first semester including metric spaces and then arrive at chapter 11 . Several possibilities for things to cover after metric spaces, depending on time are: 1) 8.1 –8.5 , 10.1 –10.5 , 10.7 (multivariable calculus, focus on multivariable integral). 2) Chapter 8 , chapter 9 , 10.1 and 10.2 (multivariable calculus, focus on path integrals). 3) Chapters 8, 9, and 10 (multivariable calculus, path integrals, multivariable integrals). 4) Chapters 8, (maybe 9 ), and 11 (multivariable differential calculus, some advanced analysis). 5) Chapter 8 , chapter 9 , 11.1, 11.6, 11.7 (a simpler variation of the above). 6 INTRODUCTION Chapter 8 Several Variables and Partial Derivatives 8.1 Vector spaces, linear mappings, and convexity Note: 2–3 lectures 8.1.1 Vector spaces The euclidean space Rn has already made an appearance in the metric space chapter. In this chapter, we extend the differential calculus we created for one variable to several variables. The key idea in differential calculus is to approximate functions linear functions (approximating the graph by a straight line). In several variables, we must introduce a little bit of linear algebra before we can move on. We start with vector spaces and linear mappings on vector spaces. While it is common to use~v or the bold v for elements of Rn, especially in the applied sciences, we use just plain old v, which is common in mathematics. That is, v Rn is a vector, which means 2 v = (v1;v2;:::;vn) is an n-tuple of real numbers.* It is common to write and treat vectors as column vectors, that is, n-by-1 matrices: " v1 # v2 v = (v ;v ;:::;v ) = . 1 2 n . vn We will do so when convenient. We call real numbers scalars to distinguish them from vectors. We often think of vectors as a direction and a magnitude and draw the vector as an arrow. The vector (v1;v2;:::;vn) is represented by an arrow from the origin to the point (v1;v2;:::;vn), see 2 Figure 8.1 in the plane R . When we think of vectors as arrows, they are not based at the origin necessarily; a vector is simply the direction and the magnitude, and it does not know where it starts. On the other hand, each vector also represents a point in Rn. Usually, we think of v Rn as 2 a point if we are thinking of Rn as a metric space, and we think of it as an arrow if we think of the so-called vector space structure on Rn (addition and scalar multiplication). Let us define the abstract notion of a vector space, as there are many other vector spaces than just Rn. *Subscripts are used for many purposes, so sometimes we may have several vectors that may also be identified by subscript, such as a finite or infinite sequence of vectors y1;y2;:::. 8 CHAPTER 8. SEVERAL VARIABLES AND PARTIAL DERIVATIVES x2 (v1;v2) x1 Figure 8.1: Vector as an arrow. Definition 8.1.1. Let X be a set together with the operations of addition, +: X X X, and × ! multiplication, : R X X, (we usually write ax instead of a x). X is called a vector space (or a · × ! · real vector space) if the following conditions are satisfied: (i) (Addition is associative) If u;v;w X, then u + (v + w) = (u + v) + w. 2 (ii) (Addition is commutative) If u;v X, then u + v = v + u. 2 (iii) (Additive identity) There is a 0 X such that v + 0 = v for all v X. 2 2 (iv) (Additive inverse) For every v X, there is a v X, such that v + ( v) = 0. 2 − 2 − (v) (Distributive law) If a R, u;v X, then a(u + v) = au + av. 2 2 (vi) (Distributive law) If a;b R, v X, then (a + b)v = av + bv. 2 2 (vii) (Multiplication is associative) If a;b R, v X, then (ab)v = a(bv). 2 2 (viii) (Multiplicative identity) 1v = v for all v X. 2 Elements of a vector space are usually called vectors, even if they are not elements of Rn (vectors in the “traditional” sense). If Y X is a subset that is a vector space itself using the same operations, then Y is called a ⊂ subspace or a vector subspace of X. Multiplication by scalars works as one would expect. For example, 2v = (1 + 1)v = 1v + 1v = v + v, similarly 3v = v + v + v, and so on. One particular fact we often use is that 0v = 0, where the zero on the left is 0 R and the zero on the right is 0 X. To see this start with 2 2 0v = (0 + 0)v = 0v + 0v, and add (0v) to both sides to obtain 0 = 0v. Similarly v = ( 1)v, − − − which follows by ( 1)v + v = ( 1)v + 1v = ( 1 + 1)v = 0v = 0.

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