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Contents 1. Review of Sets and Functions 4 1.1. Unions, Intersections, Complements, and Products 5 1.2. Functions 6 1.3. Finite

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Contents 1. Review of Sets and Functions 4 1.1. Unions, Intersections, Complements, and Products 5 1.2. Functions 6 1.3. Finite Contents 1. Review of Sets and Functions4 1.1. Unions, intersections, complements, and products5 1.2. Functions6 1.3. finite and countable sets8 1.4. Worked Examples 10 1.5. Exercises 10 1.6. Problems 11 2. The Real Numbers 11 2.1. The least upper bound property 12 2.2. Archimedean properties and square roots 13 2.3. Exercises 14 2.4. Worked Example 15 2.5. Problems 15 3. Metric Spaces 16 3.1. Definitions and Examples 16 3.2. Normed vector spaces 18 3.3. Open Sets 20 3.4. Closed Sets 23 3.5. The interior, closure, and boundary of a set 25 3.6. Exercises 26 3.7. Worked Examples 27 3.8. Problems 27 4. Sequences 30 4.1. Definitions and examples 30 4.2. Sequences and closed sets 32 4.3. Monotone real sequences 32 4.4. Limit theorems 35 4.5. Subsequences 37 4.6. The limits superior and inferior 38 4.7. Worked Examples 40 4.8. Exercises 41 4.9. Problems 42 1 2 5. Cauchy sequences and completeness 45 5.1. Worked Examples 47 5.2. Exercises 48 5.3. Problems 48 6. Compact Sets 49 6.1. Definitions and Examples 49 6.2. Compactness and closed sets 51 6.3. Sequential Compactness 53 6.4. The Heine-Borel theorem 54 6.5. Worked Example 55 6.6. Exercises 55 6.7. Problems 56 7. Connected sets 58 7.1. Exercises 60 7.2. Problems 60 8. Continuous Functions 61 8.1. Definitions and Examples 61 8.2. Continuity and Limits 63 8.3. Continuity of Rational Operations 64 8.4. Continuity and Compactness 65 8.5. Uniform Continuity and Compactness 66 8.6. Continuity and Connectedness 67 8.7. The Completion of a Metric Space - optional 68 8.8. Exercises 69 8.9. Problems 70 9. Sequences of Functions and the Metric Space C(X) 73 9.1. Sequences of Functions 73 9.2. The Metric Space C(X) 75 9.3. Exercises 76 9.4. Problems 77 10. Differentiation 78 10.1. Definitions and Examples 78 10.2. Basic Properties 79 3 10.3. The Mean Value Theorem 81 10.4. Taylor's Theorem 83 10.5. Vector-Valued Functions 84 10.6. Exercises 84 10.7. Problems 84 11. Riemann Integration 87 11.1. Definition of the Integral 87 11.2. Sufficient Conditions for Integrability 90 11.3. Properties of the Integral 93 11.4. The Lp norms 94 11.5. Integration and Differentiation 95 11.6. Integration of vector valued functions 97 11.7. Differentiability of a limit 99 11.8. Exercises 99 11.9. Problems 100 12. Series 101 12.1. Some Numerical Sequences 102 12.2. Numerical Series 103 12.3. The Root Test 106 12.4. Series Squibs 107 12.5. Power Series 109 12.6. Functions as Power Series 111 12.7. Taylor Series 112 12.8. Exercises 113 12.9. Problems 115 13. Complex Numbers and Series∗ 119 13.1. Complex Numbers 119 13.2. Power Series and Complex Numbers 120 13.3. Dirichlet Series 122 13.4. Problems 125 14. Linear Algebra Review 126 14.1. Matrices and Linear Maps Between Euclidean Spaces 126 14.2. Norms on Rn 129 4 14.3. The vector space of m × n matrices 131 14.4. The set of invertible matrices 132 14.5. Exercises 133 14.6. Problems 133 15. Derivatives of Mappings Between Euclidean Spaces 135 15.1. The Derivative: definition and examples 135 15.2. Properties of the Derivative 136 15.3. Directional Derivatives 137 15.4. Partial derivatives and the derivative 139 15.5. Exercises 141 15.6. Problems 142 16. The Inverse and Implicit Function Theorems 143 16.1. Lipschitz Continuity 143 16.2. The Inverse Function Theorem 144 16.3. The Implicit Function Theorem 147 16.4. Immersions, Embeddings, and Surfaces∗ 150 16.5. Exercises 152 16.6. Problems 152 1. Review of Sets and Functions It is assumed that the reader is familiar with the most basic set constructions and functions and knows the natural numbers, integers and rational numbers N =f0; 1; 2;::: g Z =f0; ±1; ±2;::: g m =f : m 2 ; n 2 +g Q n Z N respectively as well as the real numbers R, though we will carefully review the least upper bound property of R in Section2. Let N+ = N n f0g denote the positive integers. Familiarity with matrices Mn(R) and Mm;n(R) is also assumed. The complex numbers C will also appear in these course notes. 5 1.1. Unions, intersections, complements, and products. Definition 1.1. Given sets X; Y ⊂ S, the union and intersection of X and Y are X [ Y =fz 2 S : z 2 X or z 2 Y g ⊂ S X \ Y =fz 2 S : z 2 X and z 2 Y g ⊂ S; respectively. The complement of X, denoted X;~ is the set X~ = fx 2 S : x2 = Xg: The relative complement of X in Y is Y n X = Y \ X~ = fz 2 S : z 2 Y and z2 = Xg: In particular, X~ = S n X. / Definition 1.2. Let X and Y be sets. The Cartesian product of X and Y is the set X × Y = f(x; y): x 2 X; y 2 Y g: / Example 1.3. The set R2 = R × R is known as the Cartesian plane. The set R3 is the 3-dimensional Euclidean space of third semester Calculus. 4 Definition 1.4. Given a set S, let P (S) denote the power set of S, the set of all subsets of S. / Example 1.5. Let S = f0; 1g. Then, P (S) = f;; f0g; f1g; f0; 1gg: As we shall see later, P (N) is a large set. 4 Definition 1.6. Let S be a given set. The union and intersection of the collection F ⊂ P (S) are [A2F A =fx 2 S : there is a A 2 F such that x 2 Ag \A2F A =fx 2 S : x 2 A for every A 2 Fg: respectively. / For example, let F = fA ⊂ N : 0 2 Ag. In this case, \A2F = f0g and [A2F = N: Do Problem 1.1. 6 1.2. Functions. Definition 1.7. A function f is a triple (f; A; B) where A and B are sets and f is a rule which assigns to each a 2 A a unique b = f(a) in B. We write f : A ! B: (i) The set A is the domain of f. (ii) The set B is the codomain of f (iii) The range of f, sometimes denoted rg(f), is the set ff(a): a 2 Ag: (iv) The function f : A ! B is one-one if x; y 2 A and x 6= y implies f(x) 6= f(y). (v) The function f : A ! B is onto if for each b 2 B there exists an a 2 A such that b = f(a); i.e., if rg(f) = B. (vi) A function which is both one-one and onto is called a bijection. (vii) The graph of f is the set graph(f) = f(a; f(a)) : a 2 Ag ⊂ A × B: (viii) If f : A ! B and Y ⊂ B, the inverse image of Y under f is the set f −1(Y ) = fx 2 A : f(x) 2 Y g: (ix) If f : A ! B and C ⊂ A, the set f(C) =ff(c): c 2 Cg =fb 2 B : there is an c 2 C such that b = f(c)g is the image of C under f. Note, rg(f) = f(A). (x) The identity function on a set A is the function idA : A ! A with rule idA(x) = x. / Example 1.8. Often one sees functions specified by giving the rule only, leaving the domain implicitly understood (and the codomain unspecified), a practice to be avoided. For example, given f(x) = x2 it is left to the reader to guess that the domain is the set of real numbers. But it could also be C or even Mn(C), the n × n matrices with entries from C. If the domain is taken to be R, then R is a reasonable choice of codomain. However, the range of f is [0; 1) (a fact which will be carefully proved later) and so the codomain could be any set containing [0; 1). The moral is that it is important to specify both the domain and codomain as well as the rule when defining a function. 4 Example 1.9. Define f : R ! R by f(x) = x2. Note that f is neither one-one nor onto. As an illustration of the notion of inverse image, f −1((4; 1)) = (−∞; 2) [ (2; 1) and f −1((−2; −1)) = ;. 4 Example 1.10. The function g : R ! [0; 1) defined by g(x) = x2 is not one-one, but it is, as we'll see in Subsection 2.2, onto. The function h : [0; 1) ! [0; 1) defined by h(x) = x2 is both one-one and onto. Note h−1((4; 1)) = (2; 1). 4 7 Do Exercise 1.1 and Problem 1.2. Remark 1.11. Given a set S; an (index) set I and a function α : I ! P (S), let Ai = α(i). Let F denote the range of α. In this context, the union and intersection over F are written, [i2I Ai = [A2F A and \i2I Ai = \A2F A respectively. See Exercise 1.3. Finally, in the case that the index set is N or Z, it is customary to write, 1 [n2NAn = [n=0An and 1 [n2ZAn = [n=−∞An: Definition 1.12.
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