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Mathematical Analysis Mathematical Analysis by Tom Lindstrøm Department of Mathematics University of Oslo 2014 i Preface These notes were first written as an emergency measure when the textbook for the course MAT2400 failed to show up in the spring of 2011. The present version is corrected, updated and extended, but still far from perfect. I would like to thank everybody who has pointed out mistakes and weaknesses in the previous versions, in particular Geir Ellingsrud, Erik Løw, Per Størset, and Daniel Aubert. Blindern, January 7th, 2014 Tom Lindstrøm ii Contents 1 Preliminaries: Proofs, Sets, and Functions 3 1.1 Proofs . 3 1.2 Sets and boolean operations . 6 1.3 Families of sets . 9 1.4 Functions . 11 1.5 Relations and partitions . 13 1.6 Countability . 16 2 Metric Spaces 21 2.1 Definitions and examples . 22 2.2 Convergence and continuity . 27 2.3 Open and closed sets . 31 2.4 Complete spaces . 38 2.5 Compact sets . 41 2.6 An alternative description of compactness . 47 2.7 The completion of a metric space . 50 3 Spaces of continuous functions 57 3.1 Modes of continuity . 57 3.2 Modes of convergence . 59 3.3 The spaces C(X; Y ) ....................... 63 3.4 Applications to differential equations . 66 m 3.5 Compact subsets of C(X; R ).................. 70 3.6 Differential equations revisited . 75 3.7 Polynomials are dense in C([a; b]; R) . 80 3.8 Baire's Category Theorem . 86 4 Series of functions 93 4.1 lim sup and lim inf . 93 4.2 Integrating and differentiating sequences . 95 4.3 Power series . 101 4.4 Abel's Theorem . 104 4.5 Normed spaces . 108 iii CONTENTS 1 4.6 Inner product spaces . 113 4.7 Linear operators . 124 5 Measure and integration 129 5.1 Measure spaces . 130 5.2 Complete measures . 139 5.3 Measurable functions . 144 5.4 Integration of simple functions . 150 5.5 Integrals of nonnegative functions . 157 5.6 Integrable functions . 167 5.7 L1(X; A; µ) and L2(X; A; µ) . 173 6 Constructing measures 181 6.1 Outer measure . 182 6.2 Measurable sets . 184 6.3 Carath´eodory's Theorem . 189 6.4 Lebesgue measure on R . 196 6.5 Approximation results . 200 6.6 The coin tossing measure . 204 6.7 Product measures . 207 6.8 Fubini's Theorem . 210 2 CONTENTS Chapter 1 Preliminaries: Proofs, Sets, and Functions Chapters with the word "preliminaries" in the title are never much fun, but they are useful | they provide the reader with the background information necessary to enjoy the main body of the text. This chapter is no exception, but I have tried to keep it short and to the point; everything you find here will be needed at some stage, and most of the material will show up throughout the book. Mathematical analysis is a continuation of calculus, but it is more ab- stract and therefore in need of a larger vocabulary and more precisely defined concepts. You have undoubtedly dealt with proofs, sets, and functions in your previous mathematics courses, but probably in a rather casual way. Now they become the centerpiece of the theory, and there is no way to un- derstand what is going on if you don't have a good grasp of them: The subject matter is so abstract that you can no longer rely on drawings and intuition; you simply have to be able to understand the concepts and to read, create and write proofs. Fortunately, this is not as difficult as it may sound if you have never tried to take proofs and formal definitions seriously before. 1.1 Proofs There is nothing mysterious about mathematical proofs; they are just chains of logically irrefutable arguments that bring you from things you already know to whatever you want to prove. Still there are a few tricks of the trade that are useful to know about. Many mathematical statements are of the form \If A, then B". This simply means that whenever statement A holds, statement B also holds, but not necessarily vice versa. A typical example is: "If n 2 N is divisible by 14, then n is divisible by 7". This is a true statement since any natural 3 4CHAPTER 1. PRELIMINARIES: PROOFS, SETS, AND FUNCTIONS number that is divisible by 14, is also divisible by 7. The opposite statement is not true as there are numbers that are divisible by 7, but not by 14 (e.g. 7 and 21). Instead of \If A, then B", we often say that \A implies B" and write A =) B. As already observed, A =) B and B =) A mean two different things. If they are both true, A and B hold in exactly the same cases, and we say that A and B are equivalent. In words, we say \A if and only if B", and in symbols we write A () B. A typical example is: \A triangle is equilateral if and only if all three angels are 60◦" When we want to prove that A () B, it is often convenient to prove A =) B and B =) A separately. If you think a little, you will realize that \A =) B" and \not-B =) not-A" mean exactly the same thing | they both say that whenever A happens, so does B. This means that instead of proving \A =) B", we might just a well prove \not-B =) not-A". This is called a contrapositive proof, and is convenient when the hypothesis \not-B" gives us more to work on than the hypothesis \A". Here is a typical example. Proposition 1.1.1 If n2 is an even number, so is n. Proof: We prove the contrapositive statement: "If n is odd, so is n2": If n is odd, it can be written as n = 2k + 1 for a nonnegative integer k. But then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 which is clearly odd. 2 It should be clear why a contrapositive proof is best in this case: The hy- pothesis \n is odd" is much easier to work with than the original hypothesis \n2 is even". A related method of proof is proof by contradiction or reductio ad ab- surdum. In these proofs, we assume the opposite of what we want to show, and prove that this leads to a contradiction. Hence our assumption must be false, and the original claim is established. Here is a well-known example. p Proposition 1.1.2 2 is an irrational number. p Proof: We assume for contradiction that 2 is rational. This means that p m 2 = n for natural numbers m and n. By cancelling as much as possible, we may assume that m and n have no common factors. 1.1. PROOFS 5 If we square the equality above and multiply by n2 on both sides, we get 2n2 = m2 This means that m2 is even, and by the previous proposition, so is m. Hence m = 2k for some natural number k, and if we substitute this into the last formula above and cancel a factor 2, we see that n2 = 2k2 This means that n2 is even, and by the previous proposition n is even. Thus we have proved that both m and n are even, which is impossible as we as- sumedp that they have no common factors. This means thatp the assumption that 2 is rational leads to a contradiction, and hence 2 must be irra- tional. 2 Let me end this section by reminding you of a technique you have cer- tainly seen before, proof by induction. We use this technique when we want to prove that a certain statement P (n) holds for all natural num- bers n = 1; 2; 3;:::. A typical statement one may want to prove in this way, is n(n + 1) P (n) : 1 + 2 + 3 + ··· + n = 2 The basic observation behind the technique is: 1.1.3 (Induction Principle) Assume that P (n) is a statement about nat- ural numbers n = 1; 2; 3;:::. Assume that the following two conditions are satisfied: (i) P (1) is true (ii) If P (k) is true for a natural number k, then P (k + 1) is also true. Then P (n) holds for all natural numbers n. Let us see how we can use the principle to prove that n(n + 1) P (n) : 1 + 2 + 3 + ··· + n = 2 holds for all natural numbers n. First we check that the statement holds for n = 1: In this case the formula says 1 · (1 + 1) 1 = 2 which is obviously true. Assume now that P (k) holds for some natural number k, i.e. k(k + 1) 1 + 2 + 3 + ··· + k = 2 6CHAPTER 1. PRELIMINARIES: PROOFS, SETS, AND FUNCTIONS We then have k(k + 1) (k + 1)(k + 2) 1 + 2 + 3 + ··· + k + (k + 1) = + (k + 1) = 2 2 which means that P (k + 1) is true. By the Induction Principle, P (n) holds for all natural numbers n. Exercises for Section 1.1 1. Assume that the product of two integers x and y is even. Show that at least one of the numbers is even. 2. Assume that the sum of two integers x and y is even. Show that x and y are either both even or both odd. 2 3. Show that if n is a natural number suchp that n is divisible by 3, then n is divisible by 3. Use this to show that 3 is irrational. 1.2 Sets and boolean operations In the systematic development of mathematics, set is usually taken as the fundamental notion from which all other concepts are developed.
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