real analysis; brief lecture notes contentsastrombe/reellanalys2018/lecturenotes.pdf · real...

REAL ANALYSIS; BRIEF LECTURE NOTES

Contents

1. Definitions and properties of real numbers 22. Metric spaces 33. Metric spaces; compactness 54. Limits 75. Subsequences; Cauchy sequences; limsup and liminf 86. Banach’s fixed point theorem; series 97. Continuity 108. Continuity (II) 119. Differentiation 1210. Differentiation (II) 1311. The Riemann Integral 1412. The Riemann Integral (II) 1513. Sequences and series of functions 1714. Sequences and series of functions (II) 1815. Sequences and series of functions (III) 1916. Sequences and series of functions (IV) 2017. Functions of several variables 2118. Functions of several variables (II) 2219. The Inverse Function Theorem and the Implicit Function Theorem 2320. The Rank Theorem, and derivatives of higher order 24

1

2 REAL ANALYSIS; BRIEF LECTURE NOTES

1. Definitions and properties of real numbers

In this lecture I will discuss material from Chapter 1 of the book, withparticular focus on the sections “Ordered Sets” (1.5–1.11), “Fields” (1.12–1.18), “the Real Field” (1.19–1.22), and “the Extended Real Number Sys-tem” (1.23).

Note: Paragraph 1.3 on p. 3 contains some definitions from basic settheory; please check that you know these. Please also check that you arecomfortable with the summation symbol notation, e.g. “

∑11k=5(k

2−3k)” and

“∑b

j=a

∑j2+5k=j2

ek+j”. This notation is introduced in Paragraph 1.34 on p. 15

(for complex numbers, but of course as a special case it applies when theterms are real numbers).

Suggested problems:

Exercises 4,5,7 in Ch. 1.

Exam 2014-04-23: problem 1.

L 1.1. Let A and B be nonempty sets of real numbers, and assume thata ≤ b for all a ∈ A and b ∈ B. Prove that supA and inf B are real numbers(i.e. not ±∞), and that supA ≤ inf B.

(Exercise 6 in Ch. 1 is also relevant for the course, but quite difficult!Also all the other exercises in Ch. 1 seem to be “good”; however they mainlyaddress stuff that you should know from previous courses, rather than stuffthat are central in the present course – except exercise 20.)

REAL ANALYSIS; BRIEF LECTURE NOTES 3

2. Metric spaces

I will start by saying some brief words about the remaining parts of Chap-ter 1: “The Complex Field” (note in particular the Schwarz inequality; The-orem 1.35) and “Euclidean Spaces”.

I will then turn to Chapter 2. I will go rather quickly through 2.1–2.14. Note that 2.1–2.3 contain some basic definitions about functions (=mappings = maps); please check that you are familiar with these concepts! Iwill often use surjective as a synonym of “onto”, and injective as a synonymof “1-1”, and bijective as a synonym of “1-1 and onto”. Note also that thesymbol “∼” introduced in 2.3 can be used for many things in mathematics,and it is basically only in 2.4–2.14 that we will use it to mean that twosets have the same cardinal number. Paragraph 2.7 contains basic notationfor sequences; I will typically write “(xn)” (or “(xn)

∞n=1”) for the sequence

x1, x2, x3, . . ., instead of “{xn}” as Rudin does. Also, in 2.9–2.11 Rudindiscusses basic notation regarding unions and intersections of a family ofsets; please check that you understand the notation introduced here! Inconnection with 2.9–2.11 you may also like to immediately study 2.18(g)and 2.22, since these also sort under “basic set theory”; namely 2.18(g)gives the definition of the complement, Ec, of a set E, and Theorem 2.22states that “(∪αEα)

c = ∩α(Ecα)”.

I will then turn to the main topic of this lecture: Metric Spaces. I hopeto go through Rudin’s 2.15–2.28. Here 2.15 (the definition of metric space)and 2.18 (the definition of ’neighborhood’1, ’limit point’, ’isolated point’,’closed’, ...) are important. The set Nr(p) introduced in 2.18(a) is also oftencalled an open ball (with center at p and radius r). The fact that Nr(p) isalways open is proved in Theorem 2.19.

Note that 2.17 contains some definitions which you should make sure thatyou are familiar with, although I may not spend any time on these in thelecture. Regarding the subsets of R defined in 2.17; (a, b), [a, b], [a, b) and(a, b] I will call all of these intervals; in particular I prefer to call “(a, b)”an (open) interval, instead of “segment” as Rudin does. I may often alsouse standard notation for unbounded intervals, such as (a,∞) and (−∞, b].2

Thus, in the (standard) notation which I will use, there are the following

1The definition of “neighborhood” in 2.18(a) is non-standard. In most books, a “neigh-borhood of a point p” is defined to be any subset (alt: open subset) V ⊂ X which contains

Nr(p) for some r > 0. However in this course we will stick to Rudin’s definition of “neigh-borhood”; please remind me if you note that I forget this! You may also note that Rudinlater, in Def. 4.32, introduces a special new concept called a “neighborhood of +∞ (resp.,of −∞)” (in R).

2Rudin introduces “(a,∞)” and “(−∞, a)” in Def. 4.32.


types of intervals (here a, b ∈ R):

[a, b] := {x ∈ R : a ≤ x ≤ b}

[a, b) := {x ∈ R : a ≤ x < b}

(a, b] := {x ∈ R : a < x ≤ b}

(a, b) := {x ∈ R : a < x < b}

(−∞, a] = {x ∈ R : x ≤ a}

(−∞, a) = {x ∈ R : x < a}

[a,∞) = {x ∈ R : x ≥ a}

(a,∞) = {x ∈ R : x > a}.

Here the first four intervals are bounded and the last four are unbounded(cf. 2.18(i)). Also [a, b] and (−∞, a] and [a,∞) are closed while (a, b) and(−∞, a) and (a,∞) are open. Later one may also note that using the abovenotation, we have that a subset of R is connected if and only if it is aninterval! (It may be a nice exercise to prove this fact, after you have studiedTheorem 2.47.)

More about Rudin’s 2.17: Regarding the concept “open ball (with centerx and radius r)”, note that this is exactly the set Nr(x) defined in 2.18(a),in the special case of X = R

k. The remaining concepts in 2.17, i.e. “k-cell”and “convex”: I suggest that you digest these already now, although I willcome back to them later.

Suggested problems:

Exercises 5,6,10,11 in Ch. 2.

(Also the exercises 7,8,9 in Rudin’s Ch. 2 are highly relevant for the course,and exercises 1–4 are ’fairly relevant’.)

L 2.1. Let E be a bounded subset of a metric space (X, d). Prove that forevery point q ∈ X there is a real number M such that d(p, q) < M for allp ∈ E.

L2.2. In Rudin’s exercise 2.11, for each of those dj which are metrics on R,prove that a subset of R is open with respect to dj if an only if it is open withrespect to the standard metric on R. Also give an example of a metric onR which does not yield exactly the same open sets as the standard metric.


3. Metric spaces; compactness

In this lecture I hope to cover 2.29–2.42. Thus I will continue discussingmetric spaces, and in particular the very important notion of a set beingcompact, which is defined in 2.32.

Regarding 2.38–2.42: The big goal in those paragraphs is to prove theimportant Theorem 2.41, which describes what it means to be a compactset in R

k. I will start by discussing the statement of Theorem 2.41 (note inparticular that (a)⇔(b) is the famous Heine-Borel Theorem; also (b)⇔(c)is true in any metric space X, not just in R

k). I will then discuss the proofof Theorem 2.41, following 2.38–2.41.

A remark regarding Theorem 2.38: Since I use different (today standard)conventions regarding “intervals” than Rudin does (cf. Lecture #2), I willformulate Theorem 2.38 as follows: “If (In) is a sequence of closed andbounded intervals in R, such that In ⊃ In+1 (n = 1, 2, 3, . . .), then ∩∞

1 In isnot empty.”

Suggested problems:

Exercises 12,13,14,16,17 in Ch. 2.

(Also the all the other exercises 15,18,22–30 in Ch. 2 seem to be good andrelevant for the course.)

L3.1. Prove that every compact metric space has a countable dense subset.

[Comment: This is part of Rudin’s exercise 2.25. We will need the fact proved here later,

in the proof of the Arzela-Ascoli Theorem, 7.25.]

L3.2. Let X be the extended real number system, i.e. X = R∪{+∞,−∞}.Define d : X ×X → R by

d(x, y) =∣

∣H(x)−H(y)∣

∣,

where H : R ∪ {+∞,−∞} → R is the map given by

H(x) =

x1+|x| if x ∈ R

1 if x = +∞

−1 if x = −∞.

Prove that d is a metric on X. With this metric, prove that X is compact,and that R is an open subset of X. Prove also that the open subsets of Rwith respect to d are exactly the same as the open subsets of R with respectto the standard metric.


L 3.3. [Difficult! – and extracurricular.] Let (M,d) be a metric space. LetF (M) be the set of all non-empty compact subsets of M . We define theHausdorff distance between any two points X,Y ∈ F (M) by

dH(X,Y ) = max(

supx∈X

infy∈Y

d(x, y), supy∈Y

infx∈X

d(x, y))

.

(a) Prove that dH is a metric on F (M).(b) Prove that if M is complete then F (M) is complete.(c) Prove that if M is compact then F (M) is compact.


4. Limits

I will start the lecture by saying some brief words about the end of Chap-ter 2; 2.43–2.47. The concept of “perfect sets” is not part of the syllabus ofthe course; however I recommend that you read through 2.43–2.44 as niceexamples of concepts in metric spaces. (Note that I could certainly givea problem on the exam about “perfect sets” – but then I would state thedefinition of “perfect” in the problem formulation.) On the other hand Iconsider the notions of separated and connected to be part of the syllabusof the course, so I will discuss these a bit more (2.45–2.47).

Next, the main part of the lecture will be about convergent sequences; Iwill go through 3.1–3.4 in the book.

In connection with this Lecture #4 you may also study the paragraphs3.13, 3.14 and 3.20; these belong to the course but I may not have time todiscuss them in detail. Regarding 3.13: I often use just “increasing” as syn-onymous with “monotonically increasing”, and “decreasing” as synonymouswith “monotonically increasing”. Furthermore, we say that the sequence(sn) is strictly increasing if sn < sn+1 (n = 1, 2, 3, . . .), and strictly decreas-ing if sn > sn+1 (n = 1, 2, 3, . . .). (You may view it as a nice exercise to tryto prove Theorem 3.14 yourself. The results in Theorem 3.20 are of coursevery basic and we will use these again and again as basic tools when wecompute more complicated limits.)

Suggested problems:

Exercises 20,21 in Ch. 2. (Also exercise 19 in Ch. 2 is relevant for the course.)

Exercises 1,2,3,16 in Ch. 3. (Also exercises 17,18 in Ch. 3 are relevant forthe course.)


L 4.1. Let X be a metric space. Prove that X (“as a subset of itself”) isconnected if and only if the only sets A ⊂ X which are both open and closedare A = ∅ and A = X.

[Comment: In particular, since Rk is connected by Rudin’s exercise 2.21(c), it follows that

the only subsets of Rk which are both open and closed are Rk and ∅.]


5. Subsequences; Cauchy sequences; limsup and liminf

In this lecture I will go through 3.5–3.7 (subsequences), 3.8–3.12 (Cauchysequences) and 3.15–3.19 (lim sup and lim inf).

One detail about 3.16 (the definition of lim sup and lim inf), which I willdiscuss in the lecture: In this definition it is important to point out that wealways have E 6= ∅, so that supE and inf E are well-defined! Note that if wewould have defined E to be the set of real numbers x such that snk

→ x forsome subsequence (snk

) then it can definitely happen that E = ∅; howeversince we define E as the set of extended real numbers x such that snk

→ xfor some subsequence (snk

), we always get E 6= ∅ (proof?).

Suggested problems:

Exercises 4,5,20,23 in Ch. 3.

(Also exercises 22, 243, 25 in Ch. 3 are good and relevant for the course.)

Exam 2014-04-23: problems 2,3. Exam 2015-03–21: problem 3.

(Comment: Of course in the solution of 2014-04-23:3 you will need to assume basic prop-

erties of integrals; these should hopefully be known from previous courses, although we

will discuss how to build up a theory of integration rigorously in Lectures #11–12.)

L 5.1. Let (sn) be a sequence of real numbers. Prove that sn → +∞ (asdefined in Rudin’s 3.15) holds if and only if limn→+∞ sn = +∞ in the senseof Rudin’s 3.1 when viewing (sn) as a sequence of points in the metric spaceX = R ∪ {+∞,−∞} defined in exercise L3.2. Also prove the analogousresult for “sn → −∞”.

L 5.2. Let ℓ∞ be the space of all bounded complex sequences (an). For(an), (bn) ∈ ℓ∞ we define d((an), (bn)) = sup{|an − bn| : n ≥ 0}. Prove that(ℓ∞, d) is a metric space, and that this metric space is complete. Prove alsothat for any p ∈ ℓ∞, the “closed unit ball”, E = {q ∈ ℓ∞ : d(q, p) ≤ 1} is abounded and closed set in ℓ∞, but not compact.

Hint: One way to prove that E is not compact is to construct a sequence of points in E

which has no convergent subsequence.

3Exercise 24(b) has a slight misprint in my opinion; I think that the last sentence shouldbe replaced by, e.g.: “Show that ∆(P,Q) is well-defined, i.e. that ∆(P,Q) is unchangedif (pn) and (qn) are replaced by equivalent sequences. Prove also that ∆ is a distancefunction in X∗.”


6. Banach’s fixed point theorem; series

I will start by proving Banach’s Fixed Point Theorem (= the contractionprinciple) from Rudin’s 9.22–9.23; this result can be understood already nowand is a nice example on Cauchy sequences and completeness.

I will next discuss series. Here I will discuss the definition and basic prop-erties rather carefully; 3.21–3.25. In connection with 3.25 one can alreadymention the concepts of absolute and non-absolute convergence; cf. 3.45,3.46. Next, paragraphs 3.26–3.37 are about series with nonnegative terms.(Such series are particularly important to study since we face such a se-ries whenever we ask if an arbitrary series is absolutely convergent.) Pleasereview 3.26 yourself. I will discuss 3.27 and the general idea of dyadic de-composition; 3.28–3.29 can be seen as simple examples of that idea. I willprobably not spend much or any time on 3.30–3.32, but you should studythese paragraphs and make sure that you know those results. Next, 3.33–3.36 are about the root and ratio test; you should already know about thesefrom previous courses; please read through and make sure that you knowthese facts. Theorem 3.37 (which tells us that “the root test is at leastas good as the ratio test”) can be seen as a nice exercise on lim inf andlim sup! Next, the paragraphs on power series, 3.38–3.40: You may studythese already now; however I will not discuss them now, but instead in thelater lecture about Taylor series. Next I will discuss summation by parts,3.41–3.44. I will mention the simple Theorem 3.47, but I will not talk about3.48–3.51 (about “multiplication of series”), which you can take to be ex-tracurricular. Finally I will discuss rearrangements of series, 3.52–3.55.

Suggested problems:

Exercises 6,7,8,9,10,11 in Ch. 3. (Also exercises 12,14,15,19 are relevant forthe course; however exercise 13 is extracurricular.)


L 6.1. Let ℓ1 be the space of all complex sequences (an) such that∑

anis absolutely convergent. For (an), (bn) ∈ ℓ1 we define d((an), (bn)) =∑∞

n=0 |an − bn|. Prove that (ℓ1, d) is a metric space, and that this met-ric space is complete.

L6.2. Prove that there exists a unique bounded sequence (xn)n=1,2,... of realnumbers satisfying

xn =1

n+

∞∑

m=1

xmn+ 2m

, (n = 1, 2, 3, . . .).

[Recall the space ℓ∞ which was defined in Exercise L5.2. Hint: The statement can be

proved by an application of the contraction principle in ℓ∞.]


7. Continuity

In this lecture I finished the material on series (from the last lecture) andthen discussed 4.1–4.12.

Suggested problems:

Exercises 1,2,3,4,7,8,9 in Ch. 4. (Also exercises 5,6 and 11–13 are relevantfor the course.)

L 7.1. Let X,Y,Z be metric spaces, and let subsets A ⊂ X and B ⊂ Yand maps f : A → Y and g : B → Z be given. Assume f(A) ⊂ B, sothat g ◦ f : A → Z is defined. Let p be a limit point of A and assumelimx→p f(x) = q.

(a). Assume also that q ∈ B, and that g is continuous at q. Then provethat limx→p g(f(x)) = g(q).

(b). Assume instead that q /∈ B. Prove that then q is a limit point of B.Prove also that if limy→q g(y) = u then limx→p g(f(x)) = u.

L 7.2. Let X and Y be metric spaces and assume that Y is complete. LetE ⊂ X, let f be a map from E to Y , and let p be a limit point of E.Prove that the limit limx→p f(x) exists if and only if the following holds:For every ε > 0 there is some δ > 0 such that dY (f(x), f(y)) < ε for allpoints x, y ∈ Nδ(p) \ {p}.

L7.3. LetX be a metric space, and let C(X) denote the set of all continuous,bounded functions f : X → C. For any f, g ∈ C(X) we set d(f, g) :=supx∈X |f(x) − g(x)|. Prove that (C(X), d) is a metric space. Prove alsothat this metric space is complete.

[Comment: This is carried out in 7.14–7.15 in the book; however I think it can be very

useful if you try to solve this problem already at this stage in the course!]

Exam 2014-04-23: problem 7. Exam 2015-03-21: problems 5,8.


8. Continuity (II)

The material for this lecture is 4.13–4.34 in Rudin’s book.

I may not have time to say much about 4.32–4.34 (which concern the def-inition of “limx→±∞ f(x)” and “limx→a f(x) = ±∞”), but these paragraphsare definitely part of the course.

A remark regarding Definition 4.25: Another common notation for “f(x+)”is “limt→x+ f(t)”, and similarly one can write “limt→x− f(t)” for “f(x−)”.

A remark regarding Definition 4.28: Just as for sequences (cf. Lecture #4and Rudin’s 3.13), I often use just “increasing” as synonymous with “mono-tonically increasing”, and “decreasing” as synonymous with “monotonicallyincreasing”. Furthermore, we say that a function f : (a, b) → R is strictlyincreasing if a < x < y < b implies f(x) < f(y), and strictly decreasing ifa < x < y < b implies f(x) > f(y).

Suggested problems:

Exercises 14,15,16,20,21 of Ch. 4.

(Also exercises 10,17–19,22–26 are relevant for the course.)


L8.1. Assume that ϕ is a strictly increasing continuous function that mapsan interval [A,B] onto [a, b]. Let ψ be the inverse map of ϕ. Prove that ψis a strictly increasing continuous function which maps [a, b] onto [A,B].

L8.2. Let E be a subset of R and let f be a function from E to R. Let A, x ∈R ∪ {+∞,−∞}. Prove that the following two statements are equivalent:

(I) [f(t) → A as t→ x] (as defined in Rudin’s 4.32–4.33);

(II) limt→x f(t) = A in the metric space X = R ∪ {+∞,−∞} defined inexercise L3.2, viewing E as a subset of X and f as a function from E to X.


9. Differentiation

I plan to go through 5.1–5.8 and 5.10.

A remark regarding Theorem 5.5 (the chain rule): As I will point out inthe lecture, we do not need to assume that f is continuous on [a, b] in thistheorem. (Indeed, since we are assuming that f ′(x) exists, we anyway havethat f is continuous at x, and this is all that is used in the proof.)


10. Differentiation (II)

I plan to go through 5.9 and 5.11–5.19.

Suggested problems (for #9 and #10):

Exercises 1,5,8,9,16,26 in Ch. 5. In exercise 26, consider also the case whenf is vector-valued.

(Also all the other exercises in Ch. 5 are relevant for the course.)


L10.1. (A slightly generalized version of the problem which I posed duringLecture #10.) Suppose that f is a real valued function defined in a neigh-borhood of x ∈ R, and suppose that f ′(x) exists. Let a, b ∈ R. Compute

limh→0

f(x+ ah)− f(x+ bh)

h.


11. The Riemann Integral

I plan to go through 6.1–6.11.

Some comments regarding the definition of a partition of [a, b], in Rudin’s6.1: I think that it is a bit confusing that although Rudin defines a partitionof [a, b] to be a finite set of points in [a, b], he still allows equality (i.e., allowsxi = xi+1) in “a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b”. I will not allow this inmy lectures. Specifically, I will formulate the definition as follows:

A partition of [a, b] is a finite subset P ⊂ [a, b] satisfying a, b ∈ P . Weoften write “P = (xi)

ni=0”, by which we mean that P = {x0, x1, . . . , xn}

and a = x0 < x1 < · · · < xn = b. We then also write ∆xi := xi − xi−1

for i = 1, . . . , n.

It should also be noted that this use of the word “partition” is different from the

standard notion of “partition” from basic theory of sets and equivalence relations (which

you are hopefully all well aware of). Namely, according to that standard definition, a

“partition of [a, b]” means instead: “a family F of subsets of [a, b] such that ∪S∈FS = [a, b]

and S ∩ S′ = ∅ for any S 6= S′ ∈ F”.4

NOTE: Rudin in his Chapter 6 defines also the more general Riemann-

Stieltjes integral, “∫ b

af dα =

∫ b

af(x) dα(x)”. This concept is extracurricular

for our course. (However I recommend that you learn about it anyway, sinceyou may find this useful later in life.) The Riemann integral is the specialcase “α(x) ≡ x” of the Riemann-Stieltjes integral.

4However the two different notions are not too far from each other: Namely toany partition P = (xi)

n0 of [a, b] one can associate the “genuine” partition (e.g.)

{[x0, x1), [x1, x2), . . . , [xn−1, xn]} of [a, b].


12. The Riemann Integral (II)

I plan to go through 6.12, 6.13 and 6.20–6.25.

(Note: There is a minor typo in Theorem 6.12: Of course we have toassume “f1 ∈ R(α) and f2 ∈ R(α)”5 also in part (b) of the theorem.)

Note that in analogy with 6.23 one also defines what it means for a complexvalued function on [a, b] to be Riemann integrable. Namely: Given a functionf : [a, b] → C we say that f is Riemann integrable on [a, b] (again written“f ∈ R”) if both Re f ∈ R and Im f ∈ R on [a, b]. When this is the case, we

define∫ b

af dx =

∫ b

a(Re f) dx+ i

∫ b

a(Im f) dx.

I will not discuss 6.14–6.19 since these paragraphs concern the Riemann-Stieltjes integral and make no sense if one only knows about the Riemannintegral. I also consider 6.26–6.27 to be extracurricular.

However, there is a special case of Theorem 6.19 which is relevant for us,which Rudin mentions in (39) on p. 133. Here is a precise formulation.

Theorem (change of variable): Suppose that f ∈ R on [a, b], and supposethat ϕ is a strictly increasing continuous function that maps an interval[A,B] onto [a, b]. Assume also that ϕ is differentiable on [A,B], and thatϕ′ ∈ R on [A,B]. Then

∫ b

a

f(x) dx =

∫ B

A

f(ϕ(y))ϕ′(y) dy.

(In particular the integral in the right hand side exists, i.e. the functiony 7→ f(ϕ(y))ϕ′(y) is Riemann integrable on [A,B].)

In the book this result follows as a consequence of two results about theRiemann-Stieltjes integral (Theorems 6.17 and 6.19). It is a nice problem toseek a direct proof; cf. Problem 12.2 below. Of course, there is an analogueof the above theorem for the case when ϕ is strictly decreasing ; and caseswhen ϕ is not monotonic can often be handled by splitting the interval [A,B]into pieces so that ϕ is monotonic on each piece.

Let us also note that if we make a bit stronger assumptions, namely as-sume that f is continuous on [a, b] and ϕ′ is continuous and positive on[A,B], then the above result can be proved quite easily using the “Fun-damental Theorem of Calculus” (i.e. Theorems 6.20, 6.21), as follows: Letψ : [a, b] → [A,B] be the inverse function of ϕ; then by Exc. 5.26, ψ is dif-ferentiable on [a, b] and ψ′(ϕ(y)) = ϕ′(y)−1 for all y ∈ [A,B]; in particular

ψ′ is continuous and positive on [a, b]. Set G(x) :=∫ ψ(x)A

f(ϕ(y))ϕ′(y) dy for

5And in our course we only care about the case when f1 ∈ R and f2 ∈ R.6With an easy addendum to handle the fact that ϕ is defined and differentiable on all

[A,B] and not just (A,B).


x ∈ [a, b]. It then follows from Theorem 6.20 and the chain rule7 that G isdifferentiable on [a, b] and

G′(x) = f(ϕ(ψ(x)))ϕ′(ψ(x))ψ(x) = f(x), ∀x ∈ [a, b].

(The last equality holds since ϕ(ψ(x)) = x and, by Exc. 5.2, ϕ′(ψ(x)) =ψ(x)−1.) Hence by Theorem 6.21,

∫ b

a

f(x) dx = G(b)−G(a) =

∫ B

A

f(ϕ(y))ϕ′(y) dy − 0,

as desired. �

Suggested problems (for #11 and #12):

Exercises 2,4,5,8,9 in Ch. 6. (Also Exercises 6,7,10–15 are relevant for thecourse.)

Exam 2014-04-23, problem 3 (again!): Prove the stronger result that∫∞π

sinxxdx

converges.

L12.1. Let a, b ∈ R, a < b. For each N ∈ Z+ we let PN be the partition

PN =

{

a+ jb− a

N: j = 0, 1, 2, . . . , N

}

of [a, b]. Prove that if f is Riemann integrable on [a, b] then

limN→∞

U(PN , f) = supN∈Z+

U(PN , f) = limN→∞

L(PN , f) = supN∈Z+

L(PN , f) =

∫ b

a

f dx.

L12.2. Prove the Change of Variable Theorem which we formulated above.

7Indeed, note that G is the composition of the function ψ : [a, b] → [A,B] and thefunction z 7→

∫z

Af(ϕ(y))ϕ′(y) dy, [A,B] → R.


13. Sequences and series of functions


Suggested problems:

Exercises 1,3,4,5,8 in Ch. 7. (Also Exercises 2,6,7,9–11 are relevant for thecourse.)


L 13.1. (Difficult!) Let I : R → R be the function given by I(x) = 0 forx irrational and I(x) = 1 for x rational. In Example 7.4, Rudin showsthat it is possible to obtain I as a “double pointwise limit” of continuousfunctions, namely I(x) = limm→∞ limn→∞(cosm!πx)2n for every x ∈ R.However, prove that it is not possible to obtain I as a single pointwise limitof continuous functions, i.e. prove that there does not exist any sequence(fn) of continuous functions from R to R satisfying limn→∞ fn(x) = I(x)for every x ∈ R.


14. Sequences and series of functions (II)

I plan to go through 7.17–7.22 and I will then state Theorem 7.25 (theArzela-Ascoli Theorem), and start discussing its proof.

Suggested problems:

Exercises 15, 16, 18 in Ch. 7. (Also Exercises 17 and 19 are relevant for thecourse.)


L 14.1. Let K be a compact metric space and let F be a subset of C(K).(We take C(K) to be equipped with the uniform metric, as always.) Provethat the closure of F is compact if and only if F is equicontinuous anduniformly bounded.8

8Definition: F is said to be uniformly bounded if ∃M ∈ R: ∀f ∈ F : ∀x ∈ K:|f(x)| < M . (I state this definition here since in Rudin’s Def. 7.19 the concept of uniformboundedness is only defined for a sequence of functions.)


15. Sequences and series of functions (III)

I will continue discussing the proof of Theorem 7.25 (this will also cover7.23, 7.24). I will then discuss the Stone–Weierstrass Theorem; 7.26–7.33.

Suggested problems:

Exercises 20 and 21 in Ch. 7. (Also Exercises 22–24 are relevant for thecourse.)



16. Sequences and series of functions (IV)

I will discuss Taylor series; I plan to go through 3.38–3.39 and 8.1–8.5.

Note that in 8.6–8.7 (also pp. 178(mid)–179), Rudin defines the exponen-tial and logarithmic and trigonometric functions, and establish their basicproperties. I will not discuss this in the lecture but you should of course beaware of these basic facts from previous courses, and you may enjoy studyingyourself how all these basic facts can now be proved in a rigorous way.

The rest of Ch. 8 is extracurricular.

Suggested problems:

Exercises 1, 5(a-b), 11 in Ch. 8. (Also Exercises 2,3,4,5(c-d),6,7,8,9 arerelevant for the course.)


17. Functions of several variables


Note that 9.1–9.5 is just basic linear algebra; please check that you alreadyknow this material. (Rudin’s definition in 9.1 of a “vector space” (over R)is very concrete since he requires any vector space to be a subset of R

k;hopefully you are aware of the standard, more abstract, definition of “vectorspace”.)


18. Functions of several variables (II)

I plan to go through 9.18–9.21 and then start discussing 9.24; the InverseFunction Theorem.

(Recall that 9.22–9.23 were already discussed in Lecture #6.)

Regarding Def. 9.20: I will write “C1-mapping”, and “f ∈ C1(E)” insteadof “C′-mapping” and “f ∈ C′(E)”.

Suggested problems for #17 and #18:

Exercises 6, 8, 10, 14 in Ch. 9. (Also exercises 7, 9, 11, 12, 13, 15, 16 arerelevant for the course – except that the statement about density in 12(d) isquite difficult to prove and can be viewed as extracurricular. Exercises 1–5concern basic linear algebra.)

L 18.1. In the proof of Theorem 9.21 (the ’difficult’ direction), prove thatit really suffices to consider the case m = 1!


19. The Inverse Function Theorem and the Implicit FunctionTheorem

I plan to go through 9.24–9.28. (Please study Example 9.29 yourself.)


20. The Rank Theorem, and derivatives of higher order

I plan to go through 9.30–9.32 and 9.39-9.41.

Suggested problems for #19 and #20:

Exercises 17, 20, 21(a), 23, 24, 26, 27 in Ch. 9. (Also exercises 18, 21(b),22, 29, 30 are relevant for the course.)

Exam 2014-04-23: problem 8. Exam 2015-03-21: problem 7.

L20.1. Let A be a linear map from Rn to R

m of rank r, let Y1 be the rangeof A and let V be an open subset of Rn. Prove that A(V ) is an open subsetof Y1. (Rudin claims that this is “clear”; cf. p. 230, line 10.)

real analysis; brief lecture notes contentsastrombe/reellanalys2018/lecturenotes.pdf · real...

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