Why Study Analysis?

aBa Mbirika

Contents

1 What is Real Analysis? 2

2 Three Reasons to Study Analysis 42.1 REASON 1: To Understand the Rigor Missing in Calculus . . . . . . . . . . . . . . 42.2 REASON 2: To Learn What Constitutes a Rigorous Mathematical Proof . . . . . . 92.3 REASON 3: The Rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Inequalities are at the HEART of Analysis! 123.1 Cauchy-Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 AM-GM Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Holder’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Two Topological Properties at the HEART of Analysis 164.1 Three Theorems at the Heart of Calculus: IVT, EVT, and UCT . . . . . . . . . . . 164.2 How the Three Theorems Depend on Connectedness and Compactness . . . . . . . 17

Section 1 WHAT IS REAL ANALYSIS? Page 2

1 What is Real Analysis?

QUESTION: What is real analysis?

ANSWER: Real analysis is a branch of mathematics that studies the behav-ior of the real numbers, sequences and series of real numbers, and real-valuedfunctions. It lays down the theoretical foundation which underlies calculus.

NOTE: This is related but distinct from the following branches:

• COMPLEX ANALYSIS: studies the behavior of the complexnumbers and complex functions.

• HARMONIC ANALYSIS: studies the behavior of harmonics(i.e., waves) such as sine waves, and how these waves synthesizeother functions via the Fourier transform.

• FUNCTIONAL ANALYSIS: studies the behavior of func-tions and how they form things like vector spaces.

Some Questions We Will Study RIGOROUSLY in RealAnalysis?

NOTE: You may already know the answer to some of these questions from lower-division classes, but in real analysis we will RIGOROUSLY understand why!!!

• Regarding Real Numbers

1. What is a real number? [Let’s discuss.]

2. Is there a largest real number?

3. What is the smallest positive real number? [Let’s discuss.]

4. How come there are “more” real numbers than rational numbers?

• Regarding Sequences and Series

5. How do you take a limit of a sequence of real numbers?

6. Which sequences have limits and which ones do not?

7. Can you add up infinitely many rational numbers that sum to a non-rational number? [Let’s discuss.]

8. If you rearrange the elements of an infinite sum, is the sum still the same?

Section 1 WHAT IS REAL ANALYSIS? Page 3

• Regarding Functions

9. What does it mean for a function be continuous? differentiable? inte-grable? bounded? [Let’s discuss.]

10. Can you “add” infinitely many functions together? And what does addi-tion mean?

11. Can you take the limit of a sequence of functions?

12. Can you differentiate and integrate an infinite series of functions?

Section 2 THREE REASONS TO STUDY ANALYSIS Page 4

2 Three Reasons to Study Analysis

2.1 REASON 1: To Understand the Rigor Missing in Calculus

QUESTION: What is the sum of the series

1 +1

2+

1

4+

1

8+

1

16+ · · · ? [You Do!]

ANSWER: Let S = 1 + 12 + 1

4 + 18 + 1

16 + · · · . Multiplying both sides by 2, weget

2S = 2 + 1 +1

2+

1

4+

1

8+

1

16+ · · ·

Hence we have 2S = 2 + S. And thus S = 2. So we have

1 +1

2+

1

4+

1

8+

1

16+ · · · = 2.

QUESTION: Using the same technique above, compute the sum of the series

1 + 2 + 4 + 8 + 16 + · · · ? [You Do!]

ANSWER: Let S = 1 + 2 + 4 + 8 + 16 + · · · . Multiplying both sides by 2, weget

2S = 2 + 1 + 2 + 4 + 8 + 16 + · · ·Hence we have 2S = S − 1. And thus S = −1. So we have

1 + 2 + 4 + 8 + 16 + · · · = −1.

which is ABSURD, obviously.

RHETORICAL QUESTION 1: Why do we trust the reasoning behindthe top answer and not the bottom answer?

Section 2 THREE REASONS TO STUDY ANALYSIS Page 5

QUESTION: Consider the following matrix and compute the following sums:[You Do!] 1 2 3

4 5 67 8 9

1. The sum of each row?

2. The sum of each column?

3. The sum of the three row sums?

4. The sum of the three column sums?

ANSWER:

1. Row 1 sum is 6, row 2 sum is 15, and row 3 sum is 24.

2. Column 1 sum is 12, column 2 sum is 15, and column 3 sum is 18.

3. The sum of the three row sums is 45.

4. The sum of the three column sums is 45.

RHETORICAL QUESTION 2: It seems that it does not matterwhether you sum the rows first or the columns first, and then add up these twoindividual sums. The answer ends up to be the same. So in series notation for thematrix A = (aij), it seems we have the following fact:

m∑i=1

n∑j=1

aij =n∑

j=1

m∑i=1

aij

where aij denotes the entry in the ith row and jth column of the matrix A. Doesthis equality hold for any two integers m and n? OBVIOUSLY, it does!!!

RHETORICAL QUESTION 3: Can we always switch the order of thesums and get the same result? For example, does the following equality hold

∞∑i=1

∞∑j=1

aij =∞∑j=1

∞∑i=1

aij ?

Section 2 THREE REASONS TO STUDY ANALYSIS Page 6

QUESTION: Consider the following matrix and compute the following sums:[You Do!]

1 0 0 0 0 · · ·−1 1 0 0 0 · · ·0 −1 1 0 0 · · ·0 0 −1 1 0 · · ·0 0 0 −1 1 · · ·...

......

...... . . .

1. The sum of each row?

2. The sum of each column?

3. The sum of the three row sums?

4. The sum of the three column sums?

ANSWER:

1. Row 1 sum is 1, row 2 sum is 0, row 3 sum is 0, etc.

2. Column 1 sum is 0, column 2 sum is 0, column 3 sum is 0, etc.

3. The sum of ALL the row sums is 1.

4. The sum of ALL the column sums is 0.

WHAT CAN WE CONCLUDE ABOUT RHETORICALQUESTION 3?: Can we always switch the order of the sums and get thesame result? For example, does the following equality hold

∞∑i=1

∞∑j=1

aij =∞∑j=1

∞∑i=1

aij ?

ANSWER: We CANNOT always interchange sums!

RHETORICAL QUESTION 4: What about switching the order ofintegrals? For example, does the following equality always hold∫ b

a

∫ d

c

f(x, y) dx dy =

∫ d

c

∫ b

a

f(x, y) dy dx ?

Section 2 THREE REASONS TO STUDY ANALYSIS Page 7

TASK 1 of 2: Compute the following double integral∫ 1

0

∫ ∞0

e−xy − xye−xy dx dy [You Do!]

ANSWER: [HINT: It may be helpful to consider the derivative of xe−xy withrespect to x.]

Observe that the derivative of xe−xy with respect to x is e−xy−xye−xy. Hence theinner integral is as follows:∫ ∞

0

e−xy − xye−xy dx = xe−xy∣∣∣∞x=0

=x

exy

∣∣∣∞x=0

=1

yexy

∣∣∣∞x=0

by L’Hopital’s Rule

= 0.

Note that in the 2nd equality we have ∞∞ since y is positive (more precisely, 0 ≤y ≤ 1). So the outer integral is simply

∫ 1

0 0 dy which is zero. Hence the doubleintegral is zero.

Section 2 THREE REASONS TO STUDY ANALYSIS Page 8

TASK 2 of 2: Compute the following double integral∫ ∞0

∫ 1

0

e−xy − xye−xy dy dx [You Do!]

ANSWER: [HINT: It may be helpful to consider the derivative of ye−xy withrespect to y.]

Observe that the derivative of ye−xy with respect to y is e−xy−xye−xy. Hence theinner integral is as follows:∫ 1

0

e−xy − xye−xy dy = ye−xy∣∣∣1y=0

= e−x − 0

= e−x.

So the outer integral is∫∞0 e−x dx = −e−x

∣∣∞x=0

= 0− (−e−0) = e0 = 1.

WHAT CAN WE CONCLUDE ABOUT RHETORICALQUESTION 4: Can we always switch the order of the integrals and get thesame result? For example, does the following equality hold∫ b

a

∫ d

c

f(x, y) dx dy =

∫ d

c

∫ b

a

f(x, y) dy dx ?

ANSWER: We CANNOT always interchange integrals!

BOTTOMLINE1:

1The person above is from the UK, hence the spelling “rigour” with the added ’u’.

Section 2 THREE REASONS TO STUDY ANALYSIS Page 9

2.2 REASON 2: To Learn What Constitutes a Rigorous MathematicalProof

Five Types of Proofs:

• (I) Direct Proofs

• (II) Proof by Cases

• (III) Proof by Contradiction

• (IV) Proof by Contrapositive

• (V) Proof by Mathematical Induction

You are expected to know how to do these five different styles of proofs fromprerequisite courses.

2.3 REASON 3: The Rewards

Here are three rewards of studying analysis:

1. Understanding the complexities of the real line

2. Understanding the many different “flavors” of convergence

3. Understanding the paradoxes of the infinite!

In order to understand the complexities of the real line, one must understandthe various subsets of it in “increasing order of sophistication”.

QUESTION: What is meant by the following sequence of set containments

N ⊆ Z ⊆ Q ⊆ R ⊆ C ⊆ GOD ?

ANSWER: The natural numbers N are contained in the integers Z, whichare contained in the rationals Q, which are contained in the reals R, which arecontained in the complex numbers C, etc..??

Section 2 THREE REASONS TO STUDY ANALYSIS Page 10

QUESTION: What are the natural numbers N?

ANSWER: These are numbers of the form

1, 2, 3, . . . .

QUESTION: What are the integers Z?

ANSWER: These are numbers of the form

. . . ,−3,−2,−1, 0, 1, 2, 3, . . . .

QUESTION: Leopold Kronecker (1823–1891) said:

“Die ganzen Zahlen hat der liebe Gott gemacht,alles andere is Menschenwerk.”

This translates in English to

“God created the natural numbers2, all else is the work of man.”

What do you think Kronecker meant by this? (NOTE: There is no CORRECTANSWER to this. So give your opinion.)

QUESTION: What is the smallest nonnegative integer?

ANSWER: 1

QUESTION: What is the smallest positive integer?

ANSWER: 0

QUESTION: Is there an integer between any two integers?

ANSWER: Sometimes. If m,n ∈ Z and n > m+ 1, then yes!

2In Kronecker’s time, the words “natural” or “whole” number were not settled yet, but most mathematiciansbelieve that “ganzen Zahlen”, which translates as “whole numbers”, most likely meant the natural numbers.

Section 2 THREE REASONS TO STUDY ANALYSIS Page 11

QUESTION: What are the rational numbers Q?

ANSWER: These are numbers in the set{

mn

∣∣∣ m,n ∈ Z and n 6= 0}.

QUESTION: What is the smallest nonnegative rational number?

ANSWER: 0

QUESTION: What is the smallest positive rational number?

ANSWER: Hmmm...

QUESTION: Is there a rational number between any two rational numbers?

ANSWER: Let p, q ∈ Q. It is clear that p+q2 is rational and between p and q.

QUESTION: Does your answer to the last question make you certain thatthere are no “gaps” in the rationals? [Let’s discuss some things!]

• Between any two real numbers, is there an integer?

• Between any two real numbers, is there a rational number?

• Between any two real numbers, is there an irrational number?

• Is there any open interval (a, b) ⊆ R that contains only rational numbers?

• Is there any open interval (a, b) ⊆ R that contains only irrational numbers?

Section 3 INEQUALITIES ARE AT THE HEART OF ANALYSIS! Page 12

3 Inequalities are at the HEART of Analysis!

MOTIVATION: A massive component of analysis is that of inequalities. Thecentral core of the classical theory of inequalities are the four results:

1. Cauchy-Schwarz Inequality

2. AM-GM (Arithmetic Mean–Geometric Mean) Inequality

3. Jensen’s Inequality, and

4. Holder’s Inequality

3.1 Cauchy-Schwarz Inequality

Cauchy is a word that we will hear 1000 times over this semester in analysis. Sowho is Cauchy??

Question 3.1. Who is Augustin-Louis Cauchy?

The following inequality is one of the most widely used and most importantinequalities in all of mathematics3.

Cauchy’s Inequality (1821)

Theorem 3.2. Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Then∣∣∣a1b1 + a2b2 + · · ·+ anbn

∣∣∣ ≤√a21 + a22 + · · ·+ a2n ·√b21 + b22 + · · ·+ b2n.

More compactly, we write∣∣∣∣∣n∑

k=0

akbk

∣∣∣∣∣ ≤(

n∑k=0

a2k

) 12

·

(n∑

k=0

b2k

) 12

.

3J. Steele: The Cauchy-Schwarz Master Class – An Introduction to the Art of Mathematical Inequalities. Cam-bridge University Press, 2004, 318 pp.

Section 3 INEQUALITIES ARE AT THE HEART OF ANALYSIS! Page 13

A Connection to Linear Algebra

QUESTION: Can you make Cauchy’s inequality∣∣∣∣∣n∑

k=0

akbk

∣∣∣∣∣ ≤(

n∑k=0

a2k

) 12

·

(n∑

k=0

b2k

) 12

on the previous page look like the so-called “Cauchy-Schwarz Identity” you weretaught in Linear Algebra?

HINT: Recall definitions of the inner product of two vectors ~a = (a1, a2, . . . , an)and ~b = (b1, b2, . . . , bn) in Rn and the norm ‖~a‖ of the vector ~a:

〈 ~a,~b 〉 :=n∑

k=0

akbk ‖~a‖ :=√a21 + a22 + · · ·+ a2n =

(n∑

k=0

a2k

) 12

.

ANSWER: The Cauchy-Schwarz inequality from Linear Algebra is∣∣∣〈 ~a,~b 〉∣∣∣ ≤ 〈 ~a, ~a 〉 12 〈~b,~b 〉 12 or equivalently∣∣∣〈 ~a,~b 〉∣∣∣ ≤ ‖~a‖ ‖~b‖.

Or maybe you learned it as ∣∣∣〈 ~a,~b 〉∣∣∣2 ≤ 〈 ~a, ~a 〉 〈~b,~b 〉.In analysis, we will prove this. ,

Hmm... Why is it called “Cauchy-Schwarz”?

Cauchy proved his inequality for sums in 1821. Bunyakovsky then proved theintegral version of it in 1859. But Schwarz gave the modern proof of it in 1889.He proved that for integrable functions f, g : [a, b]→ R, we have(∫ b

a

f(x)g(x) dx

)2

≤∫ b

a

(f(x))2 dx

∫ b

a

(g(x))2 dx.

Section 3 INEQUALITIES ARE AT THE HEART OF ANALYSIS! Page 14

3.2 AM-GM Inequality

The following identity has been described as the “oldest nontrivial inequality”(pg.285)4.

Arithmetic Mean-Geometric Mean Inequality

Theorem 3.3. Let a1, a2, . . . , an be nonnegative real number real numbers.Then

a1 + a2 + · · ·+ ann

≥ n√a1a2 · · · an.

QUESTION: Confirm the identity when a1 = 4 and a2 = 9.

ANSWER: Observe that

a1 + a22

=4 + 9

2= 6.5 ≥ 6 =

√36 =

√4 · 9 =

√a1a2.

QUESTION: How would Euclid prove AM-GM inequality for the case whenn = 2? Consider the diagram below, and prove that a+b

2 ≥√ab. [You Do!]

ANSWER: [HINT: Use the well-known geometric fact that no matter wherewe place the point D on the semi-circle, the angle ∠ADC is 90 degrees. Exploitthat fact to show 4ABD and 4DBC are similar, hence proving h =

√ab.]

Since α+ β = 90, we have γ = α and δ = β. Hence 4ABD and 4DBC have thesame corresponding angles. Thus h : a as b : h, so we have

h

a=b

h=⇒ h2 = ab =⇒ h =

√ab.

The radius a+b2 is clearly greater than or equal to h.

4J. Steele: The Cauchy-Schwarz Master Class – An Introduction to the Art of Mathematical Inequalities. Cam-bridge University Press, 2004, 318 pp.

Section 3 INEQUALITIES ARE AT THE HEART OF ANALYSIS! Page 15

A Glimpse towards Analysis II

3.3 Jensen’s Inequality

Jensen’s Inequality (1906)

Theorem 3.4. Let f : [a, b] → R be a convex function. And suppose thatp1, . . . , pn are nonnegative real numbers satifying p1 +p2 + · · ·+pn = 1. Thenwe have

f

(n∑

k=1

pkxk

)≤

n∑k=1

pk f(xk).

3.4 Holder’s Inequality

Holder’s Inequality (1889)

Theorem 3.5. Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Supposep, q ∈ (1,∞) with 1

p + 1q = 1. Then

n∑k=0

|akbk| ≤

(n∑

k=0

|ak|p) 1

p

·

(n∑

k=0

|bk|q) 1

q

.

WHY SHOULD YOU CARE?:

• Some of you might move on to Analysis II someday, or perhaps graduateschool.

• Inequalities are at the heart of analysis and are a lot of fun!

• Analysis is fun ,!

Section 4 TWO TOPOLOGICAL PROPERTIES AT THE HEART OF ANALYSIS Page 16

4 Two Topological Properties at the HEART of Analysis

4.1 Three Theorems at the Heart of Calculus: IVT, EVT, and UCT

In calculus, there are three fundamental theorems on continuous functions fromwhich the rest of calculus depends. Since you should know these already, we askyou to complete each of the statements below:

Intermediate Value Theorem (IVT)

Theorem 4.1. If f : [a, b]→ R is continuous and if r is a real number betweenf(a) and f(b), then [You Finish!]

there exists an element c ∈ [a, b] such that f(c) = r.

Extreme Value Theorem (IVT)

Theorem 4.2. If f : [a, b]→ R is continuous, then [You Finish!]

• The function f attains its maximum and minimum.

• More precisely, there exists numbers c, d ∈ [a, b] such that f(c) ≤ f(x) ≤f(d) for all x ∈ [a, b].

Uniform Continuity Theorem (UCT)

Theorem 4.3. If f : [a, b]→ R is continuous, then given ε > 0, there exists aδ > 0 such that [You Finish!]

for all x1, x2 ∈ [a, b], if |x1 − x2| < δ then |f(x1)− f(x2)| < ε.

Section 4 TWO TOPOLOGICAL PROPERTIES AT THE HEART OF ANALYSIS Page 17

QUESTION: Why are these three theorems sooooooo important in calculus?

ANSWER:

• IVT is used to construct inverse functions.

• EVT is used for proving the Mean Value Theorem for derivatives, which isthe two Fundamental Theorems of Calculus depend.

• UCT is used, among other things, to prove that every continuous function isintegrable.

4.2 How the Three Theorems Depend on Connectedness and Com-pactness

The three theorems depend not only on the continuity of the function f , but alsoon properties of the topological space [a, b].

• IVT depends on the connectedness of the space [a, b].

• EVT and UCT depend on the compactness of the space [a, b].

Both connectedness and compactness are twoimportant properties that we study in analysis.

QUESTION: What do the terms connectedness and compactness mean?[Let’s discuss informally for now.]

• Why are spaces like [a, b], (a, b], or (a, b) deemed connected?They are all one component.

• Is the space (0, 4) ∪ [4, 6] connected?Yes, since (0, 4) ∪ [4, 6] equals [0, 6].

• Is the space (0, 4) ∪ (4, 6] connected?No, it is separated into two components.

• The space [a, b] is compact, but (a, b] and (a, b) are not. What do you feelcompactness means to you?The space is bounded and contains all of its “limit points”.