N AT E S T E M E N & K E V I N Y E H R U D I N : T R A N S L AT E D

Transcription

1 N AT E S T E M E N & K E V I N Y E H R U D I N : T R A N S L AT E D

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3 Contents Preface 9 1 The Real and Complex Number System 11 Introduction 11 Ordered Sets 11 Fields 14 The Real Field 14 The Extended Real Number System 14 The Complex Field 14 Euclidean Spaces 14 Appendix 14 2 Basic Topology 15 Finite, Countable, and Uncountable Sets 15 Metric Spaces 15 Compact Sets 15 Perfect Sets 15 Connected Sets 15 3 Numerical Sequences and Series 17 Convergent Sequences 17 Subsequences 17

4 4 Cauchy Sequences 17 Upper and Lower Limits 17 Some Special Sequences 17 Series 17 Series of Nonnegative Terms 17 The Number e 17 The Root and Ratio Tests 18 Power Series 18 Summations by Parts 18 Absolute Convergence 18 Addition and Multiplication of Series 18 Rearrangements 18 4 Continuity 19 Limits of Functions 19 Continuous Functions 19 Continuity and Compactness 19 Continuity and Connectedness 19 Discontinuities 19 Monotonic Functions 19 Infinite Limits and Limits at Infinity 19 5 Differentiation 21 The Derivative of a Real Function 21 Mean Value Theorem 21 The Continuity of Derivatives 21 L Hospital s Rule 21 Derivatives of Higher Order 21 Taylor s Theorem 21 Differentiation of Vector-valued Functions 21

5 5 6 The Riemann-Stieltjes Integral 23 Definition and Existence of the Integral 23 Properties of the Integral 23 Integration and Differentiation 23 Integration of Vector-valued Functions 23 Rectifiable Curves 23 7 Sequences and Series of Functions 25 Discussion of Main Problem 25 Uniform Convergence 25 Uniform Convergence and Continuity 25 Uniform Convergence and Integration 25 Uniform Convergence and Differentiation 25 Equicontinuous Families of Functions 25 The Stone-Weierstrass Theorem 25 8 Some Special Functions 27 Power Series 27 The Exponentiation and Logarithmic Functions 27 The Trigonometric Functions 27 The Algebraic Completeness of the Complex Field 27 Fourier Series 27 The Gamma Function 27 9 Functions of Several Variables 29 Linear Transformations 29 Differentiation 29 The Contraction Principle 29 The Inverse Function Theorem 29 The Implicit Function Theorem 29

6 6 The Rank Theorem 29 Determinants 29 Derivatives of Higher Order 29 Differentiation of Integrals Integration of Differential Forms 31 Integration 31 Primitive Mappings 31 Partitions of Unity 31 Change of Variables 31 Differential Forms 31 Simplexes and Chains 31 Stokes Theorem 31 Closed Forms and Exact Forms 31 Vector Analysis The Lebesgue Theory 33 Set Functions 33 Constructions of the Lebesgue Measure 33 Measure Spaces 33 Measurable Functions 33 Simple Functions 33 Integration 33 Comparison with the Riemann Integral 33 Integration of Complex Functions 33 Functions of Class L 2 34

7 Todo list o change bullets to match with rudin o references section previous section which isn t finished yet.. 13

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9 Preface testing with some fake content

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11 1 The Real and Complex Number System Introduction Ordered Sets Often when we talk about collections of things (people, cars, dogs, etc.) we talk about how they compare to each other (height, top speed, cuteness respectively). But, we can only do this because we have a way in which these objects relate to each other. My dog is of course cuter than yours, so I might say my dog is better than yours. Symbolically, I might write this as your dog < my dog where < can be read as is less cute than in this particular scenario, but in others, it might mean has a lower top speed or really anything else you can think of. We could even take all dogs and compare them in lots of different ways, such as by weight, or tail length, or number of hairs, or... There are a lot of ways, but the idea is that with a collection of objects we often like to talk about how they relate to each other and how we can compare objects of this underlying collection or set. The following definition puts this in terms we will use through the rest of the book. Definition 1.1. If we let S be a set, then an order on S is a relation, often denoted <, with two extra properties. If x and y are in S, then only one of the following is true. x < y, x = y, y < x If x, y and z are in S and x < y and y < z, then x < z.

12 12 rudin: translated Note that we did not define what the symbol > means, but as is often done we will use it because mathematics is nothing without some abuses of notation. If we write x > y, take that to mean y < x, but instead you may read it as x is greater than y or x is larger than y. Along with this notation, we will use x y to mean that x is either less than y or it is equal to y, but we don t know which. Similarly with. While in English (and many other languages) we rely on context to understand what set and order people are using when they talk, in mathematics we have are damn pedantic. Hence the following definition. change bullets to match with rudin Definition 1.2. An ordered set is a set, together with an order defined on said set. Going back to the dogs, people often say Nate, you re dog is the cutest which would imply, if taken literally, that there is no dog that is cuter than mine. People love these kind of extremes. We have a whole book dedicated to people who are the most at something (The Guinness World Records) and it comes out every year. We also have the Olympics to find more of the most people. The fastest person on land, the fastest person in water, the fastest person on land/water/wheel (triathlon). We love this kind of thing, and of course some people are also interested in the slowest. Definition 1.3. Suppose we have an ordered set S, and a subset E S. If there exists a β S such that for every x E we have x β, then we say E is bounded above, and call β an upper bound of E. A lower bound can be defined in the same way, but with instead of. The upper and lower bounds are super helpful when discussing using relative terms. However, sometimes we wish to be more even more specific in nailing down what we are talking about and how it relates to other upper and lower bounds. Definition 1.4. Start off with taking S an ordered set, E S, and E bounded above. Additionally, there exists an α S where α is an upper bound of E. If γ < α then γ is not an upper bound of E. Then α is called the least upper bound of E. The term supremum is also used, and symbolically we write α = sup E

13 the real and complex number system 13 Similarly, we can define the greatest lower bound or infimum by taking E to be a subset of S which is bounded below, and γ > α means γ is not a lower bound. This is similarly written as α = inf E These definitions take the upper/lower bounds we defined in definition 1.3 and uses them to build a new concept which captures a measure of most which we discussed earlier. Before continuing onto more definitions lets take a look at some examples. Example 1.1. In definition 1.4 we defined a concept Definition 1.5. A set S has the least upper bound property if references section previous section which isn t finished yet for every subset E S, with E =, and E being bounded above then sup E exists and is in S. Having seen so many definitions in this section that come twofor-one, you might guess that we also have definition for greatest lower bound property. Do you see what we might change in the above definition to create the full definition? The following theorem is vitally important as it really serves as a lesson that we see all the time in mathematics; two ideas that seem more or less different, are really shown to be not so different after all. Coming across the two definitions makes the ideas feel somewhat different, but as we will see, they are indeed equivalent. Theorem 1.1. Every set with the least upper bound property, also has the greatest lower bound property, and vice-versa. Proof. Suppose S has the least upper bound property, and also that there is a non-empty subset B S that is bounded below. Define L to be the collection of all lower bounds of B (everything that is less than all of B). Symbolically, L = {x S b B, x b}. Since B is bounded below, we know L is not empty, and L is bounded above by all b B. Since L S, and S has the least upper bound property, L has a least upper bound. We write α = sup L to say α is the supremum of L. Now suppose we have a γ < α. By definition 1.4 γ cannot be a least upper bound, and since it is less than α, it must not be in B (γ / B). If anything less than α is not in B, it follows that everything

14 14 rudin: translated in B must be greater than or equal to α. That is for all b B, b α. By our definition of L above, α L. If we take a β > α, it is not in L, since α is an upper bound. We also know α is a lower bound of B since it is in L. These two facts together show us that α is the greatest lower bound of B, since anything bigger is not a lower bound. Thus the infimum exists in S, and in particular α = inf B. Thus the least upper bound property implies the greatest lower bound property. Proofing the other way round is nearly the same argument with some symbols reversed. Try it out. Orders are used heavily in analysis, and more generally in all sorts of places in mathematics. They even have their own field of research called Order Theory. It s a pretty crazy place with lots of crazy things going on, and we ve only just begin touching the surface. If you enjoyed this section, go check out some more. Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix

15 2 Basic Topology Finite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets

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17 3 Numerical Sequences and Series Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e

18 18 rudin: translated The Root and Ratio Tests Power Series Summations by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements

19 4 Continuity Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity

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21 5 Differentiation The Derivative of a Real Function Mean Value Theorem The Continuity of Derivatives L Hospital s Rule Derivatives of Higher Order Taylor s Theorem Differentiation of Vector-valued Functions

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23 6 The Riemann-Stieltjes Integral Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves

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25 7 Sequences and Series of Functions Discussion of Main Problem Uniform Convergence Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone-Weierstrass Theorem

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27 8 Some Special Functions Power Series The Exponentiation and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function

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29 9 Functions of Several Variables Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order

30 30 rudin: translated Differentiation of Integrals

31 10 Integration of Differential Forms Integration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes Theorem Closed Forms and Exact Forms

32 32 rudin: translated Vector Analysis

33 11 The Lebesgue Theory Set Functions Constructions of the Lebesgue Measure Measure Spaces Measurable Functions Simple Functions Integration Comparison with the Riemann Integral Integration of Complex Functions

34 34 rudin: translated Functions of Class L 2