MATH 104 Introduction to Analysis (Fall 2004):

Instructor: Alexander Yong, 1035 Evans Hall, ayong@math.berkeley.edu

Lectures: Tuesdays and Thursdays 3:30-5:00PM, 71 Evans

Office hours: Wednesdays 2-3 PM, Fridays 3-4PM. Our GSI Norah Esty will be holding office hours on Mondays/Tuesdays in 891 Evans (10-12 PM, 1:30-4:30 PM).

Required Text: Charles Pugh, "Real Mathematical Analysis"

Other recommended texts:

1. Walter Rudin, "Principles of Mathematical Analysis"

2. Kenneth Ross, "Elementary Analysis: the theory of calculus"

Grading: Homework (25%), Midterm (25%), Final (50%)

Homework: Assignments will be posted weekly on the course webpage and are due in class the following week. The lowest two assignments will be dropped, so unfortunately, no late assignments will be accepted. All assignments will be due on Tuesdays. This is so as to coordinate with the office hours of our pool GSI.

Homework 0: Send me an email about you and your (mathematical) background so I can get to better know you.

Exams: The midterm will be Tuesday October 12th, 2004. The final exam is cumulative, with the date to be set by the registrar.

Prerequisites: Math 53, Math 54.

Syllabus: Review of elementary set theory, countable and uncountable sets, the system of real numbers and its basic properties, convergence, continuity, differentiation, integration, Euclidean spaces, metric spaces, compactness, connectedness. We will cover chapters 1-4 of Pugh, and maybe a bit of chpaters 5 and 6.


Part I: Preliminaries:

Lecture 1: Here is the first day survey (PDF file) , please fill it out and bring it to class. After some set theory notation and conventions, we started the course off by talking about the real number system. There are two fundamental questions that come to mind: (I) is our intuitive understanding enought to give a "good" mathematical definition; (II) who cares?
During the class we saw a number of attempts to deal with both. For our purposes (i.e., calculus, or "analysis") the reason that the rational numbers are insufficient is that it is not _complete_, it has too many "holes", or mathematically, there are sequences or rational number like 3, 3.1, 3.14, ... that converge, but not to anything in the rationals.
Speaking of the use of "converge" in the last sentence, isn't it circular? I mean, if we know it converges, then it must have a limit, no? Well, yes, and this lead us to the discussion about Cauchy sequences. A sequence (sorry, but I need to use the oldest calculus metaphor know to man), is about a man walking towards a wall, but take steps of length half his present distance from the wall. So he never reaches it, but the wall is his "limit". Put another way, the distances between his say, 5000th step and his 6987th step is small, and if we compare two such steps far enough along, it will be as small as we want it.
The first explanation is about a sequence with an explicit limit. The second describes a Cauchy sequence. Here's the fundamantal question: do all Cauchy sequences converge? The answer is "NO" for the rationals and "YES" for the reals. Most results about calculus/analysis trace themselves back to the need for some Cauchy sequence to converge, that is why we need to know this crucial fact about the reals, and thus need to find (different) ways to model it.
To this end, we dicussed cuts as a way to construct the reals. We then proved the least upper bound property of the reals. In fact, you'll see that this property is equivalent to the Cauchy sequence property above. Note the office hours have been changed slightly to move the Monday hour to Wednesday.

Lecture 2: We introduced Cauchy sequences, and proved the second completeness theoreem about the reals "every Cauchy sequence converges". Are the two notions of completeness logically equivalent?

In our discussions about the basics of sequences, convergence and Cauchyness we saw quite easily that if a sequence converges, it is bounded. On the other hand, not every sequence (even bounded ones) converge, as there were easy counterexamples [by the way, get used to trying to find counterexamples to statements as you read the text or come up with ideas, even if they are ultimately true, it's a useful skill to develop]. Now comes the question: does every bounded sequence have a convergent subsequence? We haven't proved this yet, but I tried to give a graphical demonstration about why it is true. Remember convergence looks something like:

X-----X-------X------XXXXXXX-----------------X----------X

while a bounded sequence looks like

[XXXXXXXXXXXXXXXXXXXX---XXXXXXX-----------XXXXXXXXXXXXXXX]

OK, the limits of ASCII are obvious, but the point is that if **you** take a pen, and try to draw a sequence on a line of length, say 5 inches, sooner or later, you're going to find yourself having to draw a "fat" point, no matter how hard you try. This **intuitively** explains that the answer is yes (why??). WARNING: do NOT regard this as a proof! OK, if it's not a proof, then why talk about it at all? The point is that do keep organized, we must at least **believe** the statement first (and **forget about the proof(!?)**), at some intuitive level. These diagrams are useful for that, even if ultimately, they do not belong in a proof that you show to your friends.

I'm away this Friday, please feel free to drop by Tuesday at 11AM for "make-up" office hours.

Lecture 3: Assignment 1: pg 40 of Pugh, Questions 11, 12(a), 16, 19, 28, 35(a), 38; due next Tuesday (or if you like, here is the .pdf version ). (Bonus problem 1: prove that every bounded sequence has a convergent subsequence; Bonus problem 2: prove that on the earth, at any instant, there are two antipodal points with the exact same temperature and barometric pressure. You may make continuity assumptions). We talked about existence theorems and the difference between a constructive and nonconstructive solution. We moved on to Euclidean space, dot products, norms, the Cauchy-Schwarz identity, triangle inequality and generalizations to inner product spaces.

Lecture 4: More on inner product spaces. We then talked about bijections, and cardinality. We showed that the reals are uncountable.

Lecture 5: Assignment 2: Chapter 1: Q 20, 26, 27, 30(a), 32, 38; due next Tuesday. We finished off the discussion of Cardinality. We proved the Extreme value theorem. The basic idea is encoded in the definition of V_x and looking at the lub of X.

Lecture 6: We finished Chapter 1 by proving the intermediate value theorem (which is similar to the proof of the extreme value theorem, in style). We started our study of general metric spaces and got as far as talking about homeomorphisms.

Lecture 7: Assignment 3: Chapter 2: 1, 5, 11, 22, 26.
We'll talk about the _topology_ of metric spaces, i.e., what are open sets in a metric space and how do they interact with one another. The topology way of thinking about metric spaces (versus the sequence way ) can be particularly useful and elegant at times. Prime example: continuity of a function can be reformulated as "inverse images of open sets are open" (as opposed to the delta epison definition).

Lecture 8: We proved the topological definition of continuity of a function. We considered the following problem: show that a function f:M->N such that any sequence {a_n} converges iff {f(a_n)} converges implies f is continuous. We talked about some more metric spaces: the space of square summable sequences for example. What is the distance function?

Lecture 9: Assignment 4: Chapter 2: 13, 25, 30, 34, 41, 89. We'll talk more basics about metric spaces (extending various notions from our intuition in R^m): more on closedness, boundedness, Cauchyness, completeness. The _big_ one however, will be _compactness_, perhaps the most important concept in analysis.

Lecture 10: Product metrics, completeness of R^m, boundedness. The definition of (sequential) compactness: every sequence has a convergent subsequence. Why is this definition important? How does it compare to completeness as a condition? One further hint: [a,b] is closed, but also compact and we have an extreme value theorem. But R is closed (and not compact), and we do not have an extreme value theorem.

Lecture 11: Assignment 5 (due in _two weeks_): 39, 51, 71, 85. We will continue to discuss (sequential) compactness and prove some basic properties, as well as the Bolzna-Weierstrass theorem (every bounded sequence in R^m has a convergent subsequence), and the Heine-Borel theorem (closed and bounded sets of R^m are compact). Nests of compacts; continuity and compactness. Next class will be covered by Alexander Woo, who will also have office hours (Evans 1037) that he will set. I will hold extra office hours from 10-12 noon, 1-2PM and 3-4PM Monday October 11.

Lecture 12: Alexander Woo gave an exam prep class.

Lecture 13: No lecture today. Instead we had the following midterm .

Lecture 14: We took up the midterm. The definitions (Q1) were done generally well, although there was sloppiness in a number of cases. (Q2) was not done well in general. (Q3) was intended to be easy, but many confused themselves in what was necessary to be proved, and what was part of the proof. (Q4) was also intended to be not too difficult, and it was done relatively better. Few people gave a correct proof of "yes" to Q5. Finally (Q6) was difficult. I would say that the solutions to (Q3)-(Q5) were intended to be "follow your nose" proofs. The intent was to see the examinee write down what is required for the assertion to be true, and then execute what is required. A strong example of how people went astray is (Q5). There, the problem is basically a Delta-Epsilon problem: given epsilon, find a delta (in this case the "square root of epsilon") such that the definition of continuity holds. If this is done, the problem is very easy: just check the definition of continuity in the case that f is defined as in the problem. The main point is that in this sense: _there were no tricks_. But those who misunderstood continuity or refused to do the natural thing got confused by the paradoxical nature of the function.

Afterwards we moved to uniform continuity: we proved that any continuous function on a compact set is uniformly continuous. Note that what uniform continuity says is that the "delta" does not depend on the point x. We also discussed connectedness.

Lecture 15: Assignment 6: chapter 2--54, 58, 64, 75, 77, 79. We continued our discussion of connectedness. We proved the generalized intermediate value theorem. Connectedness is a topological invariant: this makes it useful in trying to show two things are _not_ homeomorphic (others are openness, closedness, compactness...). Sometimes tricks are needed to achieve this though. The closure of a connected set is connected.

Lecture 16: The union of connected sets sharing a point is connected. Path connectedness. Path connected implies connected; the toploogist's sine curve.

Lecture 17: Assignment 7: chapter 2--101, 108, 109, 117, 118, prove that the topologist's sine curve is not path connected. Coverings and covering compactness. Equivalence of covering and sequential compactness. Total boundedness and the generalized Heine-Borel theorem.

Lecture 18: Perfect metric spaces, total disconnectedness. An interesting case study: the Cantor set.

Lecture 19: Started on Chapter 3. Asst: Chapter 3--1,2,4,5,10,13, and complete the proof of L'Hopital's rule on page 144. Differentiation, rules of differentiation, we spent some time on a careful explaination of why differentiability implies continuity (in order to illustrate the preciseness of the arguments we want), mean value theorem, ratio mean value theorem, L'Hopital's rule.

Lecture 20: The Darboux property, we tried to find (and gave) an example of a differentiable function whose derivative is not differentiable. Higher derivatives, smoothness classes. Started on analytic functions.

Lecture 21: We went over analytic functions and an example of a smooth function that is not analytic (however, we will see that all analytic funtions are smooth). Then we talked about Taylor's theorem, and proved the main result on this topic. Assignment was given in class: Chapter 3; 12, 14, 42, 50.

Lecture 22: We talked about the inverse function theorem. We started on Riemann integration. The question that we tried to understand and appreciate is "what is meant by the limit as mesh(P)->0?" The assignment was extended to next Tuesday (with some extra problems).

Lecture 23: Continued on Reimann integration, the precise definition in terms of epsilon and deltas, and which its a nice solution to the limit question. Then we compared with Darboux integrability and proved they are equivalent. Read pg 160-162 (examples).

Lecture 24: We'll deal with the Reimann-Lebesgue Theorem, which characterizes when a Reimann integral exists. A student asked (over email) why the integral of 1/x^n for n>=2 on say [0,1] exists even though there is a discontinuity at x=0. The Reimann-Lebesgue Theorem "says" why. But here is an analgoue: 1+2+3+...+n is "summable" because it's a finite sum. Now what about 1+(1/2) +(1/2)^2 + (1/2)^3 +.... (infinite sum) This sums to 2 even though it fails to be a finite sum. Although not all infinite sums converge. What is the condition for an infinite sum to converge? As you might remember, this isn't that straightforward, with a bunch of conditions which sometimes work/sometimes not: ratio test, alternating series test etc. With integration, the answer is easier (and iff). Assignment due in two weeks: Chapter 3: Q29, 46, 47, 53, 54, 59, 64.

Lecture 25: Guest lecturer Norah Esty will speak on convergence of series.

Lecture 26: Guest lecturer Norah Esty will end Chapter 3, continuing the discussion of convergence of series.

Lecture 27: We started on Chapter 4. We discussed the following question: what does it mean for a sequence of functions (f_n) to converge to a function f. There is an easy answer: pointwise convergence -- but this type of convergence fails to have a number of nice properties we want, e.g., the limit f may not be continuous even if each $f_n$ is! In contrast, we can ask for a much stronger type of convergence: uniform convergence. This has these nice properties. No assignment this week. Instead, here are some final exam review problems.

Note from Norah: I will be having one set of extra office hours next week before finals. Since the semester is technically over on Friday, I won't be able to use my usual room. However, I intend to be in South-side La Val's on Monday night from 5-8. Students should feel free to drop by if they have any questions. La Val's is a pizza place on Durant Street between Telegraph and Bowditch (about one block from campus). Be sure to say South-side La Val's, as there's also one on Euclid.