17.03.2015 Views

Studying Rudin's Principles of Mathematical Analysis Through ...

Studying Rudin's Principles of Mathematical Analysis Through ...

Studying Rudin's Principles of Mathematical Analysis Through ...

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

<strong>Studying</strong> Rudin’s <strong>Principles</strong> <strong>of</strong> <strong>Mathematical</strong> <strong>Analysis</strong> <strong>Through</strong><br />

Questions<br />

Mesut B. Çakır<br />

c○<br />

August 4, 2008


Contents<br />

1 The Real and Complex Number Systems 3<br />

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.1.1 Example: √ 2 is not rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.1.2 Remark: Gaps in Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.1.3 Definition: Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.1.4 Definition: Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.2 Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.2.1 Definition: Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.2.2 Definition: Ordered Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.2.3 Definition: Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.2.4 Definition: Supremum & Infimum . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.2.5 Examples: Bounds, Sup, and Inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.2.6 Definition: The Least-upper-bound Property . . . . . . . . . . . . . . . . . . . . . 5<br />

1.2.7 Theorem: Sup & Inf Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.3.1 Definition: Field & Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />

1.3.3 Proposition: Implications <strong>of</strong> the Addition Axioms . . . . . . . . . . . . . . . . . . 5<br />

1.3.4 Proposition: Implications <strong>of</strong> the Multiplication Axioms . . . . . . . . . . . . . . . 6<br />

1.3.5 Proposition: Implications <strong>of</strong> the Multiplication Axioms . . . . . . . . . . . . . . . 6<br />

1.3.6 Definition: Ordered Field and Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />

1.3.7 Proposition: Ordered Field Implications . . . . . . . . . . . . . . . . . . . . . . . . 6<br />

1.4 The Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />

1.4.1 Theorem: Q ⊂ R and R has the least-upper-bound property . . . . . . . . . . . . . 6<br />

1.4.2 Theorem: Archimedean Property <strong>of</strong> R and Q is Dense in R . . . . . . . . . . . . . 7<br />

1.4.3 Theorem: Uniqueness <strong>of</strong> y n = x and n√ x . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.4.4 Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.5 The Extended Real Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.5.1 Definition: Extended Real Number System . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6 The Complex Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.1 Definition: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.2 Theorem: Complex Numbers Form a Field . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.4 Definition: i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.5 Theorem: i 2 = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />

1.6.6 Theorem: (a, b) = a + bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.7 Definition: z, Re[z], Im[z] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.8 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.9 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.10 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.11 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

1.6.12 Theorem: Schwartz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

iii


iv<br />

CONTENTS<br />

1.7 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.7.1 Definition: Vector, Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.7.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.7.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8.3 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8.4 Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />

1.8.5 Step 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.8.6 Step 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.8.7 Step 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.8.8 Step 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.8.9 Step 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9 Concepts From the 2ed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.5 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />

1.9.6 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.7 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.8 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.9 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.10 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.11 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.12 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.13 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.14 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.15 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.16 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.17 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />

1.9.18 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.19 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.20 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.21 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.22 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.23 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.24 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.25 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.26 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />

1.9.27 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10 Real Numbers (2ed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.2 Theorem (Dedekind) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.6 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.10.7 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />

1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14


CONTENTS 1<br />

A Few Things to Remember When Doing <strong>Analysis</strong><br />

0. Do not make any assumptions no matter how obvious they maybe.<br />

Always start with the given definitions, axioms, and theorems.<br />

I. Pro<strong>of</strong>s and problems<br />

A. Start with the relevant axioms and definitions<br />

B. Decide what approaches fit the best<br />

1. contradiction<br />

2. straightforward pro<strong>of</strong><br />

3. induction<br />

4. brute force<br />

C. Start with the definitions stated or referred to in the question<br />

II. Topology<br />

1. look up and make sure that you understand each definition<br />

2. compare the definitions involved to what needs to be proved<br />

3. check the relevant theorems<br />

A. Solving problems involving sets (unions, intersections, etc.)<br />

1. express the elements on the LHS<br />

2. express the elements on the RHS<br />

3. manipulate either side to obtain the other side<br />

B. If having difficulty, draw a picture that may represent the situation<br />

1. lines<br />

2. rectangles, circles, etc.<br />

C. Remember that d(p, q) represents distance/radius<br />

D. Think about vectors in 2D or 3D and see if you can extend the concepts to the question in hand<br />

E. Refer to the definitions–especially the ones in 2.18 on p. 32.<br />

III. Sequences, series, sums, products<br />

A. Check which method applies or appropriate<br />

B. Tests<br />

1. ratio test<br />

2. root test<br />

3. compare to the known ones, e.g. ∑ 1<br />

n p<br />

4. remember ∑ n<br />

k=0 xk = 1−xn+1<br />

1−x<br />

obviously diverges if x ≥ 1<br />

5. see if the Schwarz inequality applies<br />

and ∑ ∞<br />

k=0 xk = 1<br />

1−x for x < 1<br />

IV. When there is nothing else to do, check the book; the start-up information needed is somewhere in<br />

there. Start with the table <strong>of</strong> contents or index.


2 CONTENTS


Chapter 1<br />

The Real and Complex Number<br />

Systems<br />

3


4 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.1 Introduction<br />

1.1.1 Example: √ 2 is not rational<br />

Show that the equation p 2 = 2 is not<br />

satisfied by any rational numbers.<br />

Hint Use contradiction. Start with<br />

the assumption that p can be expressed<br />

as a rational number.<br />

1.1.2 Remark: Gaps in Q<br />

Let A be the set <strong>of</strong> all positive rationals<br />

p such that p 2 < 2 and let B<br />

consist <strong>of</strong> all positive rationals p such<br />

that p 2 > 2. Show that A contains<br />

no largest number and B contains<br />

no smallest. In other words, show<br />

that the rational number system has<br />

certain gaps, in spite <strong>of</strong> the fact that<br />

between any two rationals there is another:<br />

If r < s then r < (r + s)/2 < s.<br />

The real number system fills these gaps.<br />

1.1.3 Definition: Set<br />

If A is any set (whose elements may be<br />

numbers or any other objects), we write<br />

x ∈ A to indicate that x is a member (or<br />

an element) <strong>of</strong> A. If x is not a member <strong>of</strong><br />

A, we write: x /∈ A. Now define empty<br />

set, nonempty, proper subset.<br />

1.1.4 Definition: Q<br />

The set <strong>of</strong> all rational numbers will be denoted by Q.<br />

1.2 Ordered Sets<br />

1.2.1 Definition: Order<br />

Let S be a set. Define order.<br />

1.2.2 Definition: Ordered Set<br />

Define an ordered set.<br />

1.2.3 Definition: Bound<br />

Define bounded above and an upper<br />

bound.


1.3. FIELDS 5<br />

1.2.4 Definition: Supremum & Infimum<br />

Define the least upper bound (supremum)<br />

and the greatest upper bound (infimum)<br />

<strong>of</strong> a set.<br />

1.2.5 Examples: Bounds, Sup, and Inf<br />

(a) Consider the sets A and B <strong>of</strong> Example<br />

1.1.1 as subsets <strong>of</strong> the ordered<br />

set Q. Comment on A and<br />

B in terms <strong>of</strong> their bounds in Q.<br />

(b) If α = sup E exists. Is it necessarily<br />

α ∈ E? Give an example.<br />

(c) Let E consist <strong>of</strong> all numbers 1/n<br />

where n=1, 2, 3, What are sup E<br />

and inf E?<br />

1.2.6 Definition: The Least-upper-bound Property<br />

Define the least-upper-bound property.<br />

1.2.7 Theorem: Sup & Inf Relation<br />

Suppose S is an ordered set with the<br />

least-uper-bound property, B ⊂ S, B<br />

is not empty, and B is bounded below.<br />

Let L be the set <strong>of</strong> all lower bounds <strong>of</strong><br />

B. Prove that α = sup L exists in S,<br />

and α = inf B, and, in particular, inf B<br />

exists in S.<br />

1.3 Fields<br />

1.3.1 Definition: Field & Field Axioms<br />

What is a field?<br />

What are the field axioms?<br />

1.3.2 Remarks<br />

(a) Is Q a field?<br />

(b) Do real and complex numbers<br />

form fields?<br />

1.3.3 Proposition: Implications <strong>of</strong> the Addition Axioms<br />

Prove that the axioms for addition imply<br />

the following statements:<br />

(a) If x + y = x + z, then y = z.<br />

(b) If x + y = x, then y = 0.<br />

(c) If x + y = 0, then y = −x.<br />

(d) −(−x) = x.


6 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.3.4 Proposition: Implications <strong>of</strong> the Multiplication Axioms<br />

Prove that the axioms for multiplication<br />

imply the following statements:<br />

(a) If x ≠ 0 and xy = xz, then y = z.<br />

(b) If x ≠ 0 and xy = x, then y = 1.<br />

(c) If x ≠ 0 and xy = 1,<br />

then y = 1/x.<br />

(d) If x ≠ 0, then 1/(1/x) = x.<br />

1.3.5 Proposition: Implications <strong>of</strong> the Multiplication Axioms<br />

Prove that the axioms for multiplication<br />

imply the following statements, for<br />

any x, y, z ∈ F.<br />

(a) 0x = 0.<br />

(b) If x ≠ 0 and y ≠ 0, then xy ≠ 1.<br />

(c) (−x)y = −(xy) = x(−y).<br />

(d) (−x)(−y) = xy.<br />

1.3.6 Definition: Ordered Field and Set<br />

Define an ordered field, ordered set.<br />

1.3.7 Proposition: Ordered Field Implications<br />

Prove that the following statements are<br />

true in every ordered field:<br />

(a) If x > 0, then −x < 0, and vice<br />

versa.<br />

(b) If x > 0 and y < z, then xy < xz.<br />

(c) If x < 0 and y < z, then xy > xz.<br />

(d) If x ≠ 0, then x 2 > 0. In particular,<br />

1 > 0.<br />

(e) If 0 < x < y, then 0 < 1/y < 1/x.<br />

Hint: Use the field axioms and the definition<br />

for an ordered field.<br />

1.4 The Real Field<br />

1.4.1 Theorem: Q ⊂ R and R has the least-upper-bound property<br />

Prove that there exists an ordered field R which has the least-upper-bound property; moreover, R contains<br />

Q as a subfield.


1.4. THE REAL FIELD 7<br />

Step 1 The members <strong>of</strong> R will be<br />

certain subsets <strong>of</strong> Q, called cuts.<br />

What three properties, by definition,<br />

does a cut (any set α) have?<br />

The letters p, q, r,... will always<br />

denote rational numbers,<br />

and α, β, γ, ... will denote cuts.<br />

Step 4 If α, β ∈ R, define α + β to<br />

be the set <strong>of</strong> all sums r + s where<br />

r ∈ α, s ∈ β. Define 0* to be the<br />

set <strong>of</strong> all negative rational numbers.<br />

It is clear that 0* is a cut<br />

(see Step 1). Verify that the axioms<br />

for addition hold in R, with<br />

0* playing the role <strong>of</strong> 0.<br />

Step 7 Define αβ, and prove the distributive<br />

law α(β +γ) = αβ +αγ.<br />

Thus completing the pro<strong>of</strong> that<br />

R is an ordered field with leastupper-bound<br />

property.<br />

Step 2 Establish R as an ordered set.<br />

Hint Define α < β.<br />

Step 5 Having proved that the addition<br />

defined in Step 4 satisfies<br />

Axioms (A) <strong>of</strong> Definition 1.3.1, it<br />

follows that Proposition 1.3.3 is<br />

valid in R. Prove one <strong>of</strong> the requirements<br />

<strong>of</strong> Definition 1.3.6: If<br />

α, β, γ in R and β < γ then,<br />

α + β < α + γ and that α > 0* if<br />

and only if −α < 0*.<br />

Step 8 Associate with each r ∈ Q the<br />

set r* which consists <strong>of</strong> all p ∈ Q<br />

such that p < r. It is clear that<br />

each r* is a cut; that is, r* ∈ R.<br />

Prove that these cuts satisfy the<br />

following relations:<br />

(a) r* + s* = (r + s)*;<br />

(b) r*s* = (rs)*;<br />

(c) r* < s* if and only if r < s<br />

Step 3 Prove that the ordered set R<br />

has the least-upper-bound property.<br />

Step 6 Multiplication is a little more<br />

bothersome than addition in the<br />

present context, since products<br />

<strong>of</strong> negative rationals are positive.<br />

For this reason we confine ourselves<br />

first to R + , the set <strong>of</strong> all<br />

α ∈ R with α > 0*.<br />

If α, β ∈ R + , define αβ to be the<br />

set <strong>of</strong> all p such that p ≤ rs for<br />

some choice <strong>of</strong> r ∈ α, s ∈ β, r ><br />

0, s > 0. Define 1* to be the set<br />

<strong>of</strong> all q < 1.<br />

Then, prove that the axioms (M)<br />

and (D) <strong>of</strong> Definition 1.12 hold,<br />

with R + in place <strong>of</strong> F and with<br />

1* in the role <strong>of</strong> 1.<br />

Step 9<br />

1.4.2 Theorem: Archimedean Property <strong>of</strong> R and Q is Dense in R<br />

Prove<br />

(a) the archimedean property <strong>of</strong> R. If<br />

x, y ∈ R, and x > 0, then there<br />

is a positive integer n such that<br />

nx > y.<br />

(b) that Q is dense in R, i.e., between<br />

any two real numbers, there is a<br />

rational one. In other words, if<br />

x, y ∈ R, and x < y, then ∃ p ∈ Q<br />

such that x < p < y.


8 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.4.3 Theorem: Uniqueness <strong>of</strong> y n = x and n√ x<br />

Prove that for every real x > 0 and every<br />

integer n > 0 there is one and only<br />

one positive real y such that y n = x.<br />

This number y is written n√ x or x 1/n .<br />

Corollary: (ab) 1/n = a 1/n b 1/n<br />

Prove that if a and b are positive real<br />

numbers and n is a positive integer,<br />

then (ab) 1/n = a 1/n b 1/n .<br />

1.4.4 Decimals<br />

Develop a system to write decimals in<br />

terms <strong>of</strong> integers by defining x = sup E.<br />

1.5 The Extended Real Number System<br />

1.5.1 Definition: Extended Real Number System<br />

What is the extended real number system?<br />

Does the extended real number<br />

system form a field? Why or why not?<br />

1.6 The Complex Field<br />

1.6.1 Definition: Complex Numbers<br />

Use an ordered pair <strong>of</strong> real numbers to<br />

define a complex number and addition<br />

and multiplication <strong>of</strong> complex numbers.<br />

1.6.2 Theorem: Complex Numbers Form a Field<br />

Use your definitions above to prove that<br />

addition and multiplication turn the set<br />

<strong>of</strong> all complex numbers into a field, with<br />

appropriate description <strong>of</strong> 0 and 1 in<br />

terms <strong>of</strong> complex numbers.<br />

1.6.3 Theorem<br />

Prove that for any real numbers a and<br />

b, we have (a, 0) + (b, 0) = (a + b, 0),<br />

and (a, 0)(b, 0) = (ab, 0).<br />

1.6.4 Definition: i<br />

Define i.<br />

1.6.5 Theorem: i 2 = −1<br />

Use the definition <strong>of</strong> i to show i 2 = −1.


1.6. THE COMPLEX FIELD 9<br />

1.6.6 Theorem: (a, b) = a + bi<br />

Prove that if a, b ∈ R, (a, b) = a + bi.<br />

1.6.7 Definition: z, Re[z], Im[z]<br />

What are<br />

Conjugate<br />

Real part,<br />

Imaginary part <strong>of</strong> z?<br />

1.6.8 Theorem<br />

If z and w are complex, prove that<br />

(a) z + w = z + w<br />

(b) zw = z · w<br />

(c) z + z = 2Re(z), z − z = 2iIm(z)<br />

(d) zz ≥ 0 and real.<br />

1.6.9 Definition<br />

What is absolute value <strong>of</strong> a complex<br />

number z?<br />

1.6.10 Theorem<br />

Let z and w be complex numbers.<br />

Prove that<br />

(a) |z| > 0 unless z = 0, |0| = 0<br />

(b) |z| = |z|<br />

(c) |zw| = |z||w|<br />

(d) |Rez| ≤ |z|<br />

(e) |z + w| ≤ |z| + |w|<br />

1.6.11 Notation<br />

What is the summation notation?<br />

1.6.12 Theorem: Schwartz Inequality<br />

What is the Schwartz inequality?<br />

State and prove it for complex numbers.


10 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.7 Euclidean Spaces<br />

1.7.1 Definition: Vector, Vector Space<br />

What are vectors, and what are their<br />

coordinates? What is a vector space<br />

over the real field, origin, null vector,<br />

norm?<br />

1.7.2 Theorem<br />

Suppose x, y, z ∈ R k<br />

Prove that<br />

and α is real.<br />

(a) |⃗x| ≥ 0;<br />

(b) |⃗x| = 0 if and only if ⃗x = 0;<br />

(c) |α⃗x| = |α||⃗x|;<br />

(d) |⃗x · ⃗y| ≤ |⃗x||⃗y|;<br />

(e) |⃗x + ⃗y| ≤ |⃗x| + |⃗y|<br />

(f) |⃗x − ⃗z| ≤ |⃗x − ⃗y| + |⃗y − ⃗z|<br />

1.7.3 Remarks<br />

Theorem 1.0.37 (a), (b), and (f) allows us to regard R k as a metric space. (see Chap. 2). R 1 is called<br />

the line; R 2 is called the plane, or the complex plane. In these two cases the norm is just the absolute<br />

value <strong>of</strong> the corresponding real or complex number.<br />

1.8 Appendix<br />

1.8.1 Step 1<br />

The members <strong>of</strong> R will be certain subsets <strong>of</strong> Q, called cuts. What three properties, by definition, does a<br />

cut (any set α) have?<br />

The letters p, q, r, will always denote rational numbers, and α, β, γ, will denote cuts.<br />

1.8.2 Step 2<br />

Establish R as an ordered set.<br />

1.8.3 Step 3<br />

Prove that the the ordered set R has the least-upper-bound property.<br />

1.8.4 Step 4<br />

If α, β ∈ R, define α + β to be the set <strong>of</strong> all sums r + s where r ∈ α, s ∈ β. Define 0* to be the set <strong>of</strong><br />

all negative rational numbers. It is clear that 0* is a cut (see Step 1). Verify that the axioms for addition<br />

hold in R, with 0* playing the role <strong>of</strong> 0.


1.9. CONCEPTS FROM THE 2ED 11<br />

1.8.5 Step 5<br />

Having proved that the addition defined in Step 4 satisfies Axioms (A) <strong>of</strong> Definition 1.12, it follows that<br />

Proposition 1.14 is valid in R. Prove one <strong>of</strong> the requirements <strong>of</strong> Definition 1.17: If α, β, γ in R and<br />

β < γ then, α + β < α + γ and that α > 0* if and only if −α < 0*.<br />

1.8.6 Step 6<br />

Multiplication is a little more bothersome than addition in the present context, since products <strong>of</strong> negative<br />

rationals are positive. For this reason we confine ourselves first to R + , the set <strong>of</strong> all α ∈ R with α > 0*.<br />

If α, β ∈ R + , define αβ to be the set <strong>of</strong> all p such that p ≤ rs for some choice <strong>of</strong> r ∈ α, s ∈ β, r ><br />

0, s > 0. Define 1* to be the set <strong>of</strong> all q < 1.<br />

Then, prove that the axioms (M) and (D) <strong>of</strong> Definition 1.12 hold, with R + in place <strong>of</strong> F and with 1*<br />

in the role <strong>of</strong> 1.<br />

1.8.7 Step 7<br />

Define αβ, and prove the distributive law α(β + γ) = αβ + αγ. Thus completing the pro<strong>of</strong> that R is an<br />

ordered field with least-upper-bound property.<br />

1.8.8 Step 8<br />

Associate with each r ∈ Q the set r* which consists <strong>of</strong> all p ∈ Q such that p < r. It is clear that each r*<br />

is a cut; that is, r* ∈ R. Prove that these cuts satisfy the following relations:<br />

(a) r* + s* = (r + s)*;<br />

(b) r*s* = (rs)*;<br />

(c) r* < s* if and only if r < s<br />

1.8.9 Step 9<br />

1.9 Concepts From the 2ed<br />

Some these definitions and theorems are covered in the appendix in the 3rd edition (see above).<br />

1.9.1 Definition<br />

What is a cut?<br />

1.9.2 Theorem<br />

If p ∈ α, and q ∉ α, prove that p < q.<br />

1.9.3 Theorem<br />

Let r be rational. Let α be the set consisting <strong>of</strong> all rational p such that p < r. Prove that α is a cut, and<br />

r is the smallest upper number <strong>of</strong> α.<br />

1.9.4 Definition<br />

What is a rational cut?<br />

1.9.5 Definition<br />

Let α and β be cuts. Define α = β and α ≠ β.


12 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.9.6 Definition<br />

Let α and β be cuts. Define α < β and α > β.<br />

1.9.7 Theorem<br />

Let α and β be cuts. Prove that either α = β or α < β or α > β.<br />

1.9.8 Theorem<br />

Let α, β, γ be cuts. Prove that if α < β and β < γ, then α < γ.<br />

1.9.9 Theorem<br />

Let α and β be cuts and γ be the set <strong>of</strong> all rationals r such that r = p + q where p ∈ α and q ∈ β. Prove<br />

that γ is a cut.<br />

1.9.10 Definition<br />

Let α and β be cuts. Define the sum α + β.<br />

1.9.11 Theorem<br />

Let α, β, γ be cuts. Prove that<br />

(a) α + β = β + α;<br />

(b) (α + β) + γ = α + (β + γ), so that parentheses may be omitted without ambiguity;<br />

(c) α + 0* = α<br />

1.9.12 Theorem<br />

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<br />

not the smallest upper number <strong>of</strong> α, and q − p = r.<br />

1.9.13 Theorem<br />

Let α be a cut. Then there is one and only one cut β such that α + β = 0*.<br />

1.9.14 Definition<br />

Define −α.<br />

1.9.15 Theorem<br />

Prove that for any cuts α, β, γ with β < γ we have α + β < α + γ; in particular, (taking β = 0*), we<br />

have α + γ > 0* if α > 0* and γ > 0*.<br />

1.9.16 Theorem<br />

Let α, β be cuts. Then there is one and only one cut γ such that α + γ = β.<br />

1.9.17 Definition<br />

Let α, β be cuts. Define β − α


1.9. CONCEPTS FROM THE 2ED 13<br />

1.9.18 Remark<br />

Group theory is not required in this book; however, those readers who are familiar with the group concept<br />

may have noticed that Theorems 1.9.9, 1.9.11,and 1.9.13 can be summarized by saying that the set <strong>of</strong><br />

cuts is a commutative group with respect to addition as defined by Definition 1.9.10.<br />

1.9.19 Theorem<br />

Let α, β be cuts such that α ≥ 0*, β ≥ 0*. Let γ consist <strong>of</strong> all negative rationals and all rational r such<br />

that r = pq where p ∈ α, q ∈ β, p ≥ 0, q ≥ 0. Prove that γ is a cut.<br />

1.9.20 Definition<br />

Let α, β be cuts. Define their product αβ.<br />

1.9.21 Definition<br />

Let α be a cut. Define its absolute value |α|.<br />

1.9.22 Definition<br />

Let α, β be cuts. Define their product αβ in terms <strong>of</strong> their absolute values.<br />

1.9.23 Theorem<br />

Let α, β be cuts. Prove that<br />

(a) αβ = βα;<br />

(b) (αβ)γ = α(βγ);<br />

(c) α(β + γ) = αβ + αγ;<br />

(d) α0* = 0*;<br />

(e) αβ = 0* only if α = 0* or β = 0*;<br />

(f) α1* = α<br />

(g) If 0* < α < β, and γ > 0* then αγ < βγ.<br />

1.9.24 Theorem<br />

Prove that if α ≠ 0*, then for every cut β, there is one and only one cut γ (which we denote by β/α)<br />

such that αγ = β.<br />

1.9.25 Theorem<br />

Prove that for any rationals p and q<br />

(a) p* + q* = (p + q)*;<br />

(b) p*q* = (pq)*;<br />

(c) p* < q* if and only if p < q<br />

1.9.26 Theorem<br />

Let α, β be cuts, and α < β. Prove that there is a rational cut r* such that α < r* < β.


14 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS<br />

1.9.27 Theorem<br />

Prove that for any cut α, p ∈ α if and only if p* < α.<br />

1.10 Real Numbers (2ed)<br />

1.10.1 Definition<br />

1.10.2 Theorem (Dedekind)<br />

Let A and B be sets <strong>of</strong> real numbers such that<br />

(a) every real number is either in A or in B;<br />

(b) no real number is in A and in B;<br />

(c) neither A nor B is empty;<br />

(d) if α ∈ A, and β ∈ B, then α < β. Prove that there is one (and only one) real number γ such that<br />

α ≤ γ for all α ∈ A, and γ ∈ B for all β ∈ B. Corollary Under the hypotheses <strong>of</strong> Theorem 1.10.2,<br />

either A contains a largest number or B contains a smallest.<br />

1.10.3 Definition<br />

Let E be a set <strong>of</strong> real numbers. What are upper and lower bounds <strong>of</strong> E.<br />

1.10.4 Definition<br />

Let E be bounded above. What the least upper bound and greatest lower bounds <strong>of</strong> E?<br />

1.10.5 Examples<br />

Let E consist <strong>of</strong> all numbers 1/n, n=1, 2, 3, What are the least upper and greatest lower bounds <strong>of</strong> E?<br />

1.10.6 Theorem<br />

Let E be a nonempty set <strong>of</strong> real numbers which is bounded above. The the lub <strong>of</strong> E exists.<br />

1.10.7 Theorem<br />

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<br />

number y is written n√ x, or x 1/n .<br />

1.11 Exercises<br />

Now, you should be ready to tackle the exercises.

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!