Mathematical Analysis Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: [email protected] Remark. You are entitled to a reward of 1 point toward a homework assignment if you are the first person to report a bona-fide mathematical mistake (i.e. not including language typos and grammatical errors.) Contents The language: Sets and mappings 4 Chapter 1. Metric spaces and continuous functions 6 1.1. Metric spaces 6 1.2. Convergent and divergent of sequences 12 1.3. Continuous functions 13 Chapter 2. Sequences and series of functions 16 2.1. Sequences and series of numbers 16 2.2. Pointwise convergence of sequences and series of functions 19 2.3. Uniform convergence of sequences and series of functions 21 2.4. Uniform convergence and continuity 22 2.5. Uniform convergence and differentiation 24 2.6. Uniform convergence and integration 25 2.7. Power series 28 2.8. Exponential function and logarithmic function 30 Chapter 3. Functions of several variables 33 3.1. Linear transformations and matrices 33 3.2. Differentiation 37 3.2.1. Partial derivatives 40 3.3. Inverse function theorem 42 3.3.1. The contraction principle 42 3.4. Implicit function theorem 46 Bibliography 47 3 The language: Sets and mappings We speak the language of sets and mappings in mathematics. Definition (Sets). • Given a set A, x 2 A denotes that x is an element (or member, point) of A and x2 = A denotes that x is not an element of A. • We say that two sets A and B are equal, denoted by A = B, if they have the same elements. • Given two sets A and B, we say that A is a subset of B, denoted by A ⊂ B (or A ⊆ B), if each element of A is an element of B; we say that A is a proper subset of B, denoted by A ( B, if A ⊆ B and A 6= B. Remark. • B ⊃ A means A ⊂ B. • Given two sets A and B, A = B if and only if (i.e. iff) A ⊂ B and B ⊂ A. Example. (1). N = f1; 2; 3; :::g denotes the set of natural numbers. (Notice that in some other books, N refers to the set f0; 1; 2; 3; :::g.) (2). Z = f:::; −3; −2; −1; 0; 1; 2; 3; :::g denotes the set of integers. (3). Q = fp=q : p; q 2 Z; q 6= 0g denotes the set of rational numbers. (4). R denotes the set of real numbers, which is a complete ordered field. (5). R n Q is called the set of irrational numbers. (6). C = fx + iy : x; y 2 Rg denotes the set of complex numbers. Definition (The empty set). The set that has no elements is called the empty set and is denoted by ;. A set that is not equal to the empty set is called nonempty. Definition (Union, intersection, and complement). Let A and B be two sets. • The union of A and B is A [ B = fx : x 2 A or x 2 Bg. • The intersection of A and B is A \ B = fx : x 2 A and x 2 Bg. We say that A and B are disjoint if A \ B = ;. • The complement of A in B is B n A = fx : x 2 B and x2 = Ag. If all the set operations are within a universal set X, then for a set A ⊂ X, we simply call X nA the complement of A, and is also denoted by Ac. Remark. • x 2 A iff x 62 Ac and x 2 Ac iff x 62 A. • E ⊂ A iff E \ Ac = ; and E ⊂ Ac iff E \ A = ;. Remark. Given a family of sets F, we define [ E = fx : x 2 E for some E 2 Fg; E2F 4 THE LANGUAGE: SETS AND MAPPINGS 5 and \ E = fx : x 2 E for all E 2 Fg: E2F In particular, if F = fEλgλ2Λ, where λ is called the index and Λ is called the index set, then we define [ Eλ = fx : x 2 Eλ for some λ 2 Λg; λ2Λ and \ Eλ = fx : x 2 Eλ for all λ 2 Λg: λ2Λ For example, n [ Ei = fx : x 2 Ei for some i = 1; :::; ng; i=1 and n \ Ei = fx : x 2 Eλ for all i = 1; :::; ng: i=1 Theorem (De Morgan's Law). Let F be a family of sets. Then !c !c [ \ \ [ E = Ec and E = Ec: E2F E2F E2F E2F Definition (Mappings). Let A and B be two sets. A mapping (or function) f from A to B, denoted by f : A ! B, is a correspondence that assigns to each element of A an element of B; for x 2 A, we denote by f(x) the assigned element in B. In the case when B = R or C, we call the mapping f a real-valued or complex-valued function. Definition (Domain and range). Let f be a mapping from A to B. We call A the domain of f. Given a subset E ⊂ A, we define f(E) = fy 2 B : y = f(x) for some x 2 Eg the image of E. We call f(A) the range of f. Definition (Inverse image). Let f be a mapping from A to B. Given a subset E ⊂ B, we define f −1(E) = fx 2 A : f(x) 2 Eg the inverse image of E. Remark. Notice that in defining the inverse images, we do not assume that f −1 is a mapping. In fact, to make f −1 a mapping, we need the following concepts. Definition (Onto and one-to-one mappings). Let f be a mapping from A to B. • We say that f is one-to-one (or injective) if x; y 2 A and x 6= y imply f(x) 6= f(y). • We say that f is onto (or surjective) if f(A) = B. Definition (Invertible mappings). Let f be a mapping from A to B. We say that f is invertible if f is both one-to-one and onto, i.e. f establishes an one-to-one correspondence (or bijection) between the sets A and B. Let f be an inverible mapping from A to B. Given each element y 2 B, there is exactly one element x 2 A such that f(x) = y and we denote by f −1(y) = x. This defines a mapping f −1 : B ! A and we call f −1 the inverse of f. CHAPTER 1 Metric spaces and continuous functions 1.1. Metric spaces Definition (Metric spaces). A metric space is a set X equipped with a function d : X × X ! [0; 1) that satisfies (i). d(p; q) = 0 iff p = q; (ii). d(p; q) = d(q; p) for all p; q 2 X; (iii). (Triangle inequality) d(p; q) ≤ d(p; r) + d(r; q) for all p; q; r 2 X. We call d a metric and d(p; q) the distance between p and q. k k Example. In R , denote a point x = (x1; :::; xk) 2 R . Then 1 2 2 2 d(x; y) = jx − yj := jx1 − y1j + ··· + jxn − ynj defines a metric on Rk, which is called the Euclidean metric. In this case, Rk is called the Euclidean space. In fact, for any 1 ≤ p ≤ 1, 1 p p p dp(x; y) = jx1 − y1j + ··· + jxn − ynj defines a metric on Rk. In particular, ρ(x; y) := d1(x; y) = sup fjxi − yijg i=1;:::;n is called the square metric. Definition. Let X be a metric space. All points and sets mentioned below are understood to be elements and subsets of X. • A neighborhood of p is a set Nr(p) consisting of all q such that d(p; q) < r, for some r > 0. The number r is called the radius of Nr(p). • A point p is an interior point of E if there is a neighborhood N of p such that N ⊂ E. E is open if every point of E is an interior point of E. • A point p is a limit point of E if every neighborhood of p contains a point q 6= p such that q 2 E. E is closed if every limit point of E is a point of E. • E is bounded if there is a real number M and a point q 2 X such that d(p; q) < M for all p 2 E. Definition (Intervals). Let a; b 2 R. We define the following intervals as subsets of R. • (a; b) = fx 2 R : a < x < bg, • (a; b] = fx 2 R : a < x ≤ bg, • [a; b) = fx 2 R : a ≤ x < bg, • [a; b] = fx 2 R : a ≤ x ≤ bg, 6 1.1. METRIC SPACES 7 • (a; 1) = fx 2 R : x > ag, • [a; 1) = fx 2 R : x ≥ ag, • (−∞; b) = fx 2 R : x < bg, • (−∞; b] = fx 2 R : x ≤ bg, Remark. In Rudin, only [a; b] are referred as \intervals"; while (a; b) are called \segments". Definition (Cells). Let ai; bi 2 R and ai < bi for i = 1; :::; k. We define a k-cell fx = (x1; :::; xk): ai ≤ xi ≤ bi for all i = 1; :::; kg = [a1; b1] × · · · × [ak; bk] as a subset of Rk. In particular, a 1-cell is an interval and a 2-cell is a rectangle. Remark. • If 0 < r < s, then Nr(p) ⊂ Ns(p). k • The neighborhood Nr(p) of p in the Euclidean space R is also called the ball with center p and radius r.