Math 321 Real Analysis

Math 321 Real Analysis David PerkinsonContents Foreword Chapter 1. i Metric Spaces1 §1.Equivalence Relations1 §2.Cardinality I2 §3.Cardinality II3 §4.Metric Spaces4 §5.Closed Sets and Limits5 §6.Continuity7 §7.Miscellaneous8 §8.Completeness9 §9.Completeness II11 §10.Picard’s Theorem12 Chapter 2. Topological Spaces 15 §1.Topology15 §2.Separation Axioms16 §3.Connectedness17 §4.Compactness I18 §5.Compactness In Metric Spaces19 §6.Compactness in Metric Spaces II20 §7.Compactness in Rn21 Chapter 3. Valuations and Completions 23 §1.Valuations23 §2.Classification24 §3.Completions26 Chapter 4. §1. Measure Theory Measure 29 29 iiiiv Contents §2.Measure II30 §3.Measure III32 §4.Weird Sets33 §5.Weird Sets II35 Chapter 5. Integration37 §1.Measurable Functions37 §2.Integration I39 §3.Integration II40 §4.Monotone Convergence Theorem41 §5.Fatou’s Lemma43 §6.Integrable Functions44 §7.Dominated Convergence46 §8.The Riemann Integral47 §9. celebrity deathmatch: Riemann vs. Lebesgue Chapter 6. Hilbert Space 48 51 §1.Hilbert Space51 §2.L253 §3.Stone-Weierstraß54 §4.Fourier Series56 Chapter 7. Exercises 59 Index71 Bibliography75Foreword These are lecture notes from a course in real analysis first given at Reed College in the Fall of 2001. The following year, the course was repeated, but this time, each lecture was assigned to a student “scribe”. The scribe was in charge of typesetting the notes in TEX. I edited these TEX-ed notes, spliced them, and added an index. The final chapter of the notes consists of homework problems. In addition to these problems, there are exercises sprinkled throughout the text for the careful reader. By the time they get to this course, Reed students have already had an introduction to analysis (typically in the second semester) and a year-long course in multivariable calculus. These notes are an attempt to take advantage of this background. The material on metric spaces is based on I. Kaplansky’s Set Theory and Metric Spaces [3] and the material on measure theory and integration is based on R. Bartle’s The elements of integration and Lebesgue measure [1]. Other references I found useful: Topology; A First Course, [5] by J. Munkres, Principles of mathematical analysis, [6] by W. Rudin, Real analysis: modern techniques and their applications, [2] by G. Folland, and Introductory Real Analysis, [4] by A. N. Kolmogorov and S. V. Fomin. Acknowledgments. I would like to thank the Reed students that helped to make these notes: Asher Auel, Brendan Bauer-Peters, Rebecca Beachum, Ryland Bell, John Collins, Katie Davis, Rory Donovan, Rachel Epstein, Ananda Floyd, Harold Gabel, Jeffrey Green, Thomas Harlan, Alice Hill, Jeffrey Hood, Ethan Jewett, Sean Kelly, Eric Lawrence, Jeremy Levin, Michael Lieberman, Alexa Mater, Christian McNeil, Morgan Miller, Megan Othus, Duncan Ramsey, David Rosoff, John Saller, Candace Schenk, Eric Scheusner, and Nakul Shankar. Thanks also to Irena Swanson and to J. J. P. Veerman for extensive useful suggestions. iChapter 1 Metric Spaces 1. Equivalence Relations Definition 1.1. A relation on a set S is a subset R ⊆ S × S. If (a, b) ∈ R, we write a ∼R b (or just a ∼ b). Example 1.2. (1) Let S be the set of members of our class, and let (a, b) ∈ R if person a is shorter than person b. (2) Let S = Z and R = {(a, b) ∈ Z | b = 2a}. Thus, 4 ∼ 8 and 8 ∼ 16. Definition 1.3. A relation ∼ on a set S is an equivalence relation if for all a, b, c ∈ S, (1) a ∼ a (reflexivity) (2) a ∼ b =⇒ b ∼ a (symmetry) (3) a ∼ b, b ∼ c =⇒ a ∼ c (transitivity). Example 1.4. Examples of equivalence relations: (1) a ∼ b in Z if a and b have the same parity. (2) a ∼ b in Z if a − b is an integer multiple of some fixed number n. (3) p ∼ q in R2 if p − q ∈ Z2 (a donut) (4) p ∼ q on a sphere in R3 if p, q are equal or antipodal. Example 1.5. a ∼ b in Z if a = 2b is not an equivalence relation. Partitions. An equivalence relation on a set is the same thing as a partition. If ∼ is an equivalence relation on a set S, then for each a ∈ S, define the equivalence class of a to be the subset Sa := {b ∈ S | b ∼ a}. Then for all a, b ∈ S, we either have Sa = Sb or Sa ∩ Sb = ∅ (check!). Thus, the collection of distinct equivalence classes forms a partition of S. Conversely, suppose we have a partition of S. Say S = ∪α∈I Tα for some index set I, with the Tα pairwise disjoint. Define a ∼ b in S if a, b ∈ Tα for some α. Then ∼ is an equivalence relation on S (check!). 12 1. Metric Spaces 2. Cardinality I Definition 2.1. Two sets S and T have the same cardinality if there is a bijection f : S → T . Proposition 2.2. Let F be a collection of sets. For S, T ∈ F, say S ∼ T if S and T have the same cardinality. Then ∼ is an equivalence relation on F. Proof. For reflexivity, use the identity map; for symmetry, the inverse of bijection is bijec- tion; and for transitivity, a composition of bijections is bijection.  Definition 2.3. A set is countably infinite if it has the same cardinality as N = {1, 2, 3, . . . }. A set is countable if it is countably infinite or finite. Proposition 2.4. A subset of a countable set is countable. Proof. We may assume S is a subset of N. By the well-ordering principle, S has a smallest element, say s1 . Then S {s1 } has a smallest element s2 , etc. We get a list of all elements of S: s1 s2 s3 . . . .  Theorem 2.5. A countable union of countable sets is countable. Proof. Let S1 , S2 , . . . be a list of countable sets. List the elements of each Si : si1 , si2 , si3 . . . (If Si is finite with k elements, define sij = sik for j ≥ k) List the Si ’s in an array s11 s12 s13 · · · s21 s22 s23 · · · s31 s32 s33 · · · .. .. .. . . . . . . Then list the elements in the array by reading off the diagonals: s11 , s12 , s21 , s13 , s22 , s31 , s14 , s23 , . . . Throw out repeats to get a list of elements of ∪Si .  Corollary 2.6. Z is countable. Proof. Z = {. . . , −2, −1} ∪ {0} ∪ {1, 2, . . . }.  Corollary 2.7. Q is countable. S Proof. For each n ∈ N, define Qn = { na ∈ Q | a ∈ Z}. Then Q = n∈N Qn and each Qn is countable (easy bijection to Z).  Definition 2.8. α ∈ C is algebraic if there exists f ∈ Z[x] such that f (α) = 0. Theorem 2.9. The collection A of algebraic numbers is countable. Proof. For each f ∈ Z[x], let roots(f ) = {α ∈ C | f (α) = 0}. Then roots(f ) is finite for all f . We can write A = ∪f ∈Z[x] roots(f ), so it suffices to show that Z[x] is countable. Note Z[x] = ∪d≥0 Z[x]d where Z[x]d is in the collection of polynomials in Z[x] with degree d. So it is sufficient to show Z[x]d is countable for each d. For each f ∈ Z[x] where f= d X i=0 ai xi3. Cardinality II 3 define m(f ) := d X |ai |. i=0 Then Z[x]d = ∪k≥0 {f ∈ Z[x]d | m(f ) = k} This exhibits Z[x]d as a countable union of finite sets. Hence, Z[x]d is countable.  Theorem 2.10 (Cantor 1892, first proof 1874). R is not countable. Proof. It suffices to show some subset of R is uncountable. Let T be all non-negative reals less than 1 with decimal expansion involving only 0s and 1s. Suppose T is countable; say f : N → T is a bijection. Define x ∈ T by x = 0.x1 x2 x3 . . . where xi is 0 if the i-th digit of f (i) is 1, else xi = 1. Then x differs from f (i) in the i-th digit for all i. Thus, x is not in the image of f , contradicting the fact that f is a bijection.  3. Cardinality II Theorem 3.1. R is uncountable. Corollary 3.2. Transcendental (i.e., nonalgebraic) real numbers exist. Proof. Real algebraic numbers are countable, and R is not.  In 1873, Hermite showed that e is transcendental, and in 1882, Lindemann showed π is transcendental. Hilbert’s seventh problem was √to show that αβ is transcendental if α, β are algebraic, α 6= 0, 1, and β is irrational. (Eg., 7 2 , ii = e−π/2 ). This was proved by Gelfond in 1934. Definition 3.3. Let S and T be sets. We’ll write (1) |S| = |T | if S and T have the same cardinality, (2) |S| ≤ |T | if there is an injection S ,→ T , and (3) |S| |T | if |S| ≤ |T | and there does not exist an injection T ,→ S. Facts: • If |S| ≤ |T | and |T | ≤ |S|, then |S| = |T | (Cantor-Bernstein Theorem). • If |S| ≤ |T | and |T | ≤ |U |, then |S| ≤ |U |. • For all S, T , either |S| ≤ |T | or |T | ≤ |S|. Notation: If S is a set, then 2S is the set of all subsets of S. Note: If S is finite, we’ll identify |S| with a natural number in the usual way. Then if S is finite, |2S | = 2|S| . Theorem 3.4. For every set S, |S| |2S |. Proof. First note that |S| ≤ |2S | using the natural injection, S → 2S x 7→ {x}4 1. Metric Spaces We’ll now show there is no surjection f : S  2S . Suppose there is. Define B = {s ∈ S | s ∈ / f (s)}. Since f is surjective, there exists b ∈ S such that f (b) = B. Question: Is b ∈ B? If yes, then b ∈ B = f (b), so b ∈ / B. If no, then b ∈ / B = f (b), so b ∈ B. This is a contradiction.  What if we had defined B = {s ∈ S | s ∈ f (s)}? Russell’s Paradox. Let U be the set of all sets. How could we have |U| |2U |? The problem here is that U is not a set. (To adapt the argument we used above to this specific situation, consider the set B = {A ∈ U | A ∈ / A}. Is B ∈ B? Albeit it strange, it is possible for a set to be an element of itself. Can you think of an example?) Fact: |2N | = |R|. Can you prove this? Continuum Hypothesis (Hilbert’s First Problem). Is there a set S such that |N| |S| |R|? The continuum hypothesis (CP) is that there is no such S. In 1938, K. Gödel showed that CP is consistent with the ZF axioms, then in 1963, P. Cohen showed that the negation of CP is consistent with the ZF axioms. So CP is independent of the ZF axioms. We’re free to accept or reject it as a new axiom. 4. Metric Spaces Definition 4.1. A metric space is a set M and a function d : M × M → R, called the metric on M , such that for all x, y, z ∈ M , (1) d(x, y) ≥ 0, and d(x, y) = 0 if and only if x = y (positive definite), (2) d(x, y) = d(y, x) (symmetry), (3) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality). Example 4.2. (1) M = R , d(x, y) = |x − y|. (2) M = Rn , d(x, y) = |x − y| = pPn 2 i=1 (xi − yi ) . (3) M = Rn , d(x, y) = max1≤i≤n {|xi − yi |}. P (4) M = Rn , d(x, y) = ni=1 |xi − yi |. (5) M = C([a, b]) = the continuous functions on [a, b] ⊆ R, d(f, g) = maxa≤t≤b |f (t) − g(t)|. (6) If (M, d) is a metric space and N ⊆ M , then (N, d) is a metric space, too (restricting d to N ). (7) Given any set M , the discrete metric on M is defined by  0 if x = y d(x, y) = 1 if x 6= y Definition 4.3. Fix a metric space (M, d). The open ball in M centered at x ∈ M with radius r 0 is the set B(x; r) = {y ∈ M | d(x, y) r}. The closed ball of radius r centered at x is {y ∈ M | d(x, y) ≤ r}.5. Closed Sets and Limits 5 Definition 4.4. A subset U ⊆ M is open if it contains an open ball about each of its points, i.e, for all u ∈ U , there exists r 0 such that B(u; r) ⊆ U . Proposition 4.5. An open ball is open. Proof. Given B(x; r), take y ∈ B(x; r). Claim: B(y, r − d(x, y)) ⊆ B(S; r). To see this, note that z ∈ B(y, r − d(x, y)) ⇒ 4-ineq. =⇒ d(y, z) ≤ r − d(x, y) ⇒ d(x, y) + d(y, z) ≤ r d(x, z) ≤ r ⇒ z ∈ B(x; r).  Proposition 4.6. (1) ∅ is open. (2) M is open (3) If {U }α∈I is any collection of open sets, then ∪α∈I Uα is open. k (4) For any k ∈ N, if U1 , . . . , Uk are open, then so is ∩ i=1 Ui . Note. Arbitrary intersections of open sets are not always open. For example, in R with the usual metric, let Un = (− n1 , n1 ). Then ∩n≥0 Un = {0}, which is not open. In fact, if arbitrary intersections of open sets are open in some metric space, then every subset of that space is open (see the homework). Definition 4.7. A neighborhood of a point x ∈ M is any set containing an open set containing x. Question: What are the open subsets of the real numbers (with usual metric)? 5. Closed Sets and Limits Let (M, d) be a metric space. Definition 5.1. Let S ⊆ M . A point x ∈ M is a contact point of S if every neighborhood of x contains a point of S. A point x ∈ M is a limit point of S if every neighborhood of x contains infinitely many points of S. A point s ∈ S is an isolated point of S if there is some neighborhood of s containing no other points of S besides s. A point s ∈ S is an interior point of S if there is a neighborhood U of s contained in S. Thus, every limit point is a contact point, and a contact point is either a limit point or an isolated point (cf. homework). Example 5.2. Let M = R with the usual metric. (1) The set of contact points of S = (0, 1) ⊂ R is [0, 1], which is the same as its set of limit points. (2) The set of contact points of S = N ⊂ R is just S, itself, and S has no limit points. Every point of S is isolated. Definition 5.3. The closure of S ⊆ M , denoted S, is the set of all contact points of S. Proposition 5.4.