Studying Rudin s Principles of Mathematical Analysis Through Questions. August 4, 2008

Transcription

1 Studying Rudin s Principles of Mathematical Analysis Through Questions Mesut B. Çakır c August 4, 2008

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3 Contents 1 The Real and Complex Number Systems Introduction Example: 2 is not rational Remark: Gaps in Q Definition: Set Definition: Q Ordered Sets Definition: Order Definition: Ordered Set Definition: Bound Definition: Supremum & Infimum Examples: Bounds, Sup, and Inf Definition: The Least-upper-bound Property Theorem: Sup & Inf Relation Fields Definition: Field & Field Axioms Remarks Proposition: Implications of the Addition Axioms Proposition: Implications of the Multiplication Axioms Proposition: Implications of the Multiplication Axioms Definition: Ordered Field and Set Proposition: Ordered Field Implications The Real Field Theorem: Q R and R has the least-upper-bound property Theorem: Archimedean Property of R and Q is Dense in R Theorem: Uniqueness of y n = x and n x Decimals The Extended Real Number System Definition: Extended Real Number System The Complex Field Definition: Complex Numbers Theorem: Complex Numbers Form a Field Theorem Definition: i Theorem: i 2 = Theorem: (a, b) = a + bi Definition: z, Re[z], Im[z] Theorem Definition Theorem Notation Theorem: Schwartz Inequality iii

4 iv CONTENTS 1.7 Euclidean Spaces Definition: Vector, Vector Space Theorem Remarks Appendix Step Step Step Step Step Step Step Step Step Concepts From the 2ed Definition Theorem Theorem Definition Definition Definition Theorem Theorem Theorem Definition Theorem Theorem Theorem Definition Theorem Theorem Definition Remark Theorem Definition Definition Definition Theorem Theorem Theorem Theorem Theorem Real Numbers (2ed) Definition Theorem (Dedekind) Definition Definition Examples Theorem Theorem Exercises

5 CONTENTS 1 A Few Things to Remember When Doing Analysis 0. Do not make any assumptions no matter how obvious they maybe. Always start with the given definitions, axioms, and theorems. I. Proofs and problems A. Start with the relevant axioms and definitions B. Decide what approaches fit the best 1. contradiction 2. straightforward proof 3. induction 4. brute force C. Start with the definitions stated or referred to in the question II. Topology 1. look up and make sure that you understand each definition 2. compare the definitions involved to what needs to be proved 3. check the relevant theorems A. Solving problems involving sets (unions, intersections, etc.) 1. express the elements on the LHS 2. express the elements on the RHS 3. manipulate either side to obtain the other side B. If having difficulty, draw a picture that may represent the situation 1. lines 2. rectangles, circles, etc. C. Remember that d(p, q) represents distance/radius D. Think about vectors in 2D or 3D and see if you can extend the concepts to the question in hand E. Refer to the definitions especially the ones in 2.18 on p. 32. III. Sequences, series, sums, products A. Check which method applies or appropriate B. Tests 1. ratio test 2. root test 3. compare to the known ones, e.g. 1 n p 4. remember n k=0 xk = 1 xn+1 1 x obviously diverges if x 1 5. see if the Schwarz inequality applies and k=0 xk = 1 1 x for x < 1 IV. When there is nothing else to do, check the book; the start-up information needed is somewhere in there. Start with the table of contents or index.

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7 Chapter 1 The Real and Complex Number Systems 3

8 4 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.1 Introduction Example: 2 is not rational Show that the equation p 2 = 2 is not satisfied by any rational numbers. Hint Use contradiction. Start with the assumption that p can be expressed as a rational number Remark: Gaps in Q Let A be the set of all positive rationals p such that p 2 < 2 and let B consist of all positive rationals p such that p 2 > 2. Show that A contains no largest number and B contains no smallest. In other words, show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If r < s then r < (r + s)/2 < s. The real number system fills these gaps Definition: Set If A is any set (whose elements may be numbers or any other objects), we write x A to indicate that x is a member (or an element) of A. If x is not a member of A, we write: x / A. Now define empty set, nonempty, proper subset Definition: Q The set of all rational numbers will be denoted by Q. 1.2 Ordered Sets Definition: Order Let S be a set. Define order Definition: Ordered Set Define an ordered set Definition: Bound Define bounded above and an upper bound.

9 1.3. FIELDS Definition: Supremum & Infimum Define the least upper bound (supremum) and the greatest upper bound (infimum) of a set Examples: Bounds, Sup, and Inf (a) Consider the sets A and B of Example as subsets of the ordered set Q. Comment on A and B in terms of their bounds in Q. (b) If α = sup E exists. Is it necessarily α E? Give an example. (c) Let E consist of all numbers 1/n where n=1, 2, 3, What are sup E and inf E? Definition: The Least-upper-bound Property Define the least-upper-bound property Theorem: Sup & Inf Relation Suppose S is an ordered set with the least-uper-bound property, B S, B is not empty, and B is bounded below. Let L be the set of all lower bounds of B. Prove that α = sup L exists in S, and α = inf B, and, in particular, inf B exists in S. 1.3 Fields Definition: Field & Field Axioms What is a field? What are the field axioms? Remarks (a) Is Q a field? (b) Do real and complex numbers form fields? Proposition: Implications of the Addition Axioms Prove that the axioms for addition imply the following statements: (a) If x + y = x + z, then y = z. (b) If x + y = x, then y = 0. (c) If x + y = 0, then y = x. (d) ( x) = x.

10 6 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS Proposition: Implications of the Multiplication Axioms Prove that the axioms for multiplication imply the following statements: (a) If x 0 and xy = xz, then y = z. (b) If x 0 and xy = x, then y = 1. (c) If x 0 and xy = 1, then y = 1/x. (d) If x 0, then 1/(1/x) = x Proposition: Implications of the Multiplication Axioms Prove that the axioms for multiplication imply the following statements, for any x, y, z F. (a) 0x = 0. (b) If x 0 and y 0, then xy 1. (c) ( x)y = (xy) = x( y). (d) ( x)( y) = xy Definition: Ordered Field and Set Define an ordered field, ordered set Proposition: Ordered Field Implications Prove that the following statements are true in every ordered field: (a) If x > 0, then x < 0, and vice versa. (b) If x > 0 and y < z, then xy < xz. (c) If x < 0 and y < z, then xy > xz. (d) If x 0, then x 2 > 0. In particular, 1 > 0. (e) If 0 < x < y, then 0 < 1/y < 1/x. Hint: Use the field axioms and the definition for an ordered field. 1.4 The Real Field Theorem: Q R and R has the least-upper-bound property Prove that there exists an ordered field R which has the least-upper-bound property; moreover, R contains Q as a subfield.

11 1.4. THE REAL FIELD 7 Step 1 The members of R will be certain subsets of Q, called cuts. What three properties, by definition, does a cut (any set α) have? The letters p, q, r,... will always denote rational numbers, and α, β, γ,... will denote cuts. Step 4 If α, β R, define α + β to be the set of all sums r + s where r α, s β. Define 0* to be the set of all negative rational numbers. It is clear that 0* is a cut (see Step 1). Verify that the axioms for addition hold in R, with 0* playing the role of 0. Step 7 Define αβ, and prove the distributive law α(β +γ) = αβ +αγ. Thus completing the proof that R is an ordered field with leastupper-bound property. Step 2 Establish R as an ordered set. Hint Define α < β. Step 5 Having proved that the addition defined in Step 4 satisfies Axioms (A) of Definition 1.3.1, it follows that Proposition is valid in R. Prove one of the requirements of Definition 1.3.6: If α, β, γ in R and β < γ then, α + β < α + γ and that α > 0* if and only if α < 0*. Step 8 Associate with each r Q the set r* which consists of all p Q such that p < r. It is clear that each r* is a cut; that is, r* R. Prove that these cuts satisfy the following relations: (a) r* + s* = (r + s)*; (b) r*s* = (rs)*; (c) r* < s* if and only if r < s Step 3 Prove that the ordered set R has the least-upper-bound property. Step 6 Multiplication is a little more bothersome than addition in the present context, since products of negative rationals are positive. For this reason we confine ourselves first to R +, the set of all α R with α > 0*. If α, β R +, define αβ to be the set of all p such that p rs for some choice of r α, s β, r > 0, s > 0. Define 1* to be the set of all q < 1. Then, prove that the axioms (M) and (D) of Definition 1.12 hold, with R + in place of F and with 1* in the role of 1. Step Theorem: Archimedean Property of R and Q is Dense in R Prove (a) the archimedean property of R. If x, y R, and x > 0, then there is a positive integer n such that nx > y. (b) that Q is dense in R, i.e., between any two real numbers, there is a rational one. In other words, if x, y R, and x < y, then p Q such that x < p < y.

12 8 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS Theorem: Uniqueness of y n = x and n x Prove that for every real x > 0 and every integer n > 0 there is one and only one positive real y such that y n = x. This number y is written n x or x 1/n. Corollary: (ab) 1/n = a 1/n b 1/n Prove that if a and b are positive real numbers and n is a positive integer, then (ab) 1/n = a 1/n b 1/n Decimals Develop a system to write decimals in terms of integers by defining x = sup E. 1.5 The Extended Real Number System Definition: Extended Real Number System What is the extended real number system? Does the extended real number system form a field? Why or why not? 1.6 The Complex Field Definition: Complex Numbers Use an ordered pair of real numbers to define a complex number and addition and multiplication of complex numbers Theorem: Complex Numbers Form a Field Use your definitions above to prove that addition and multiplication turn the set of all complex numbers into a field, with appropriate description of 0 and 1 in terms of complex numbers Theorem Prove that for any real numbers a and b, we have (a, 0) + (b, 0) = (a + b, 0), and (a, 0)(b, 0) = (ab, 0) Definition: i Define i Theorem: i 2 = 1 Use the definition of i to show i 2 = 1.

13 1.6. THE COMPLEX FIELD Theorem: (a, b) = a + bi Prove that if a, b R, (a, b) = a + bi Definition: z, Re[z], Im[z] What are Conjugate Real part, Imaginary part of z? Theorem If z and w are complex, prove that (a) z + w = z + w (b) zw = z w (c) z + z = 2Re(z), z z = 2iIm(z) (d) zz 0 and real Definition What is absolute value of a complex number z? Theorem Let z and w be complex numbers. Prove that (a) z > 0 unless z = 0, 0 = 0 (b) z = z (c) zw = z w (d) Rez z (e) z + w z + w Notation What is the summation notation? Theorem: Schwartz Inequality What is the Schwartz inequality? State and prove it for complex numbers.

14 10 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.7 Euclidean Spaces Definition: Vector, Vector Space What are vectors, and what are their coordinates? What is a vector space over the real field, origin, null vector, norm? Theorem Suppose x, y, z R k Prove that and α is real. (a) x 0; (b) x = 0 if and only if x = 0; (c) α x = α x ; (d) x y x y ; (e) x + y x + y (f) x z x y + y z Remarks Theorem (a), (b), and (f) allows us to regard R k as a metric space. (see Chap. 2). R 1 is called the line; R 2 is called the plane, or the complex plane. In these two cases the norm is just the absolute value of the corresponding real or complex number. 1.8 Appendix Step 1 The members of R will be certain subsets of Q, called cuts. What three properties, by definition, does a cut (any set α) have? The letters p, q, r, will always denote rational numbers, and α, β, γ, will denote cuts Step 2 Establish R as an ordered set Step 3 Prove that the the ordered set R has the least-upper-bound property Step 4 If α, β R, define α + β to be the set of all sums r + s where r α, s β. Define 0* to be the set of all negative rational numbers. It is clear that 0* is a cut (see Step 1). Verify that the axioms for addition hold in R, with 0* playing the role of 0.

15 1.9. CONCEPTS FROM THE 2ED Step 5 Having proved that the addition defined in Step 4 satisfies Axioms (A) of Definition 1.12, it follows that Proposition 1.14 is valid in R. Prove one of the requirements of Definition 1.17: If α, β, γ in R and β < γ then, α + β < α + γ and that α > 0* if and only if α < 0* Step 6 Multiplication is a little more bothersome than addition in the present context, since products of negative rationals are positive. For this reason we confine ourselves first to R +, the set of all α R with α > 0*. If α, β R +, define αβ to be the set of all p such that p rs for some choice of r α, s β, r > 0, s > 0. Define 1* to be the set of all q < 1. Then, prove that the axioms (M) and (D) of Definition 1.12 hold, with R + in place of F and with 1* in the role of Step 7 Define αβ, and prove the distributive law α(β + γ) = αβ + αγ. Thus completing the proof that R is an ordered field with least-upper-bound property Step 8 Associate with each r Q the set r* which consists of all p Q such that p < r. It is clear that each r* is a cut; that is, r* R. Prove that these cuts satisfy the following relations: (a) r* + s* = (r + s)*; (b) r*s* = (rs)*; (c) r* < s* if and only if r < s Step Concepts From the 2ed Some these definitions and theorems are covered in the appendix in the 3rd edition (see above) Definition What is a cut? Theorem If p α, and q α, prove that p < q Theorem Let r be rational. Let α be the set consisting of all rational p such that p < r. Prove that α is a cut, and r is the smallest upper number of α Definition What is a rational cut? Definition Let α and β be cuts. Define α = β and α β.

16 12 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS Definition Let α and β be cuts. Define α < β and α > β Theorem Let α and β be cuts. Prove that either α = β or α < β or α > β Theorem Let α, β, γ be cuts. Prove that if α < β and β < γ, then α < γ Theorem Let α and β be cuts and γ be the set of all rationals r such that r = p + q where p α and q β. Prove that γ is a cut Definition Let α and β be cuts. Define the sum α + β Theorem Let α, β, γ be cuts. Prove that (a) α + β = β + α; (b) (α + β) + γ = α + (β + γ), so that parentheses may be omitted without ambiguity; (c) α + 0* = α Theorem Let α be a cut and r > 0 be a given rationa. Then there are rationals p, q such that p α, q α, q is not the smallest upper number of α, and q p = r Theorem Let α be a cut. Then there is one and only one cut β such that α + β = 0* Definition Define α Theorem Prove that for any cuts α, β, γ with β < γ we have α + β < α + γ; in particular, (taking β = 0*), we have α + γ > 0* if α > 0* and γ > 0* Theorem Let α, β be cuts. Then there is one and only one cut γ such that α + γ = β Definition Let α, β be cuts. Define β α

17 1.9. CONCEPTS FROM THE 2ED Remark Group theory is not required in this book; however, those readers who are familiar with the group concept may have noticed that Theorems 1.9.9, ,and can be summarized by saying that the set of cuts is a commutative group with respect to addition as defined by Definition Theorem Let α, β be cuts such that α 0*, β 0*. Let γ consist of all negative rationals and all rational r such that r = pq where p α, q β, p 0, q 0. Prove that γ is a cut Definition Let α, β be cuts. Define their product αβ Definition Let α be a cut. Define its absolute value α Definition Let α, β be cuts. Define their product αβ in terms of their absolute values Theorem Let α, β be cuts. Prove that (a) αβ = βα; (b) (αβ)γ = α(βγ); (c) α(β + γ) = αβ + αγ; (d) α0* = 0*; (e) αβ = 0* only if α = 0* or β = 0*; (f) α1* = α (g) If 0* < α < β, and γ > 0* then αγ < βγ Theorem Prove that if α 0*, then for every cut β, there is one and only one cut γ (which we denote by β/α) such that αγ = β Theorem Prove that for any rationals p and q (a) p* + q* = (p + q)*; (b) p*q* = (pq)*; (c) p* < q* if and only if p < q Theorem Let α, β be cuts, and α < β. Prove that there is a rational cut r* such that α < r* < β.

18 14 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS Theorem Prove that for any cut α, p α if and only if p* < α Real Numbers (2ed) Definition Theorem (Dedekind) Let A and B be sets of real numbers such that (a) every real number is either in A or in B; (b) no real number is in A and in B; (c) neither A nor B is empty; (d) if α A, and β B, then α < β. Prove that there is one (and only one) real number γ such that α γ for all α A, and γ B for all β B. Corollary Under the hypotheses of Theorem , either A contains a largest number or B contains a smallest Definition Let E be a set of real numbers. What are upper and lower bounds of E Definition Let E be bounded above. What the least upper bound and greatest lower bounds of E? Examples Let E consist of all numbers 1/n, n=1, 2, 3, What are the least upper and greatest lower bounds of E? Theorem Let E be a nonempty set of real numbers which is bounded above. The the lub of E exists Theorem For every real x > 0 and every integer n > 0, there is one and only one real y > 0 such that y n = x. This number y is written n x, or x 1/n Exercises Now, you should be ready to tackle the exercises.