We are expect to cover a significant (but probably proper) subset of the topics below

• \(\R\) as a complete metric space and its properties
• Metric spaces
  • Cauchy sequences
  • completeness
• Topology of metric spaces
  • general topology
  • compactness
• Banach spaces
  • the space \(l^{\infty}\)
  • series and the space \(l^{1}\)
• Continuous functions on metric spaces
  • uniform continuity
  • extension of continuous functions
• Continuous functions as a Banach space
  • sequences of continuous functions
  • Ascoli-Arzelà and compact families of functions
• Differentiability and smoothness
  • differentials of functions between Banach spaces
  • Hölder continuous functions
  • Taylor expansions
• Integration
  • the Riemann–Stieltjes integral
• The contraction principle
  • the contraction principle abstractly
  • existence and uniqueness of the solution to the Cauchy problem (ODE)

Homework is an integral part of the course and represents a major part of the final grade. It will be assigned weekly. The homework is quite challenging, students are expected to work through the problems and understand the solutions that will be provided. Students are encouraged to collaborate on HW.

Expect most of your out-of-class study time to be spent on homework: the material of the course will be extensively covered by assignments (according to the idea of learning by doing). All material covered in the homework assignments is considered course material and may reappear in subsequent HW assignments, in midterms, and in the final.

Homework will be posted on this website in the corresponding section.

We will have a total of 3 exams: two midterms and a final exam. The first midterm will be graded before the drop day.

Your grade is going to be based on the homework you turn in and 3 exams. The evaluation scheme below is subject to slight modifications.

Grading scheme (subject to slight modifications)

Ask questions! Raise your (virtual) hand or if I do not notice, interrupt me! If we need to finish a concept before responding, I will tell you (with no offense intended) and address your concern as soon as we can. There are no bad questions. Let us know if you are lost trying to understand. Questions do not have to be specific to a step; examples of good questions are:

• If I had to solve this problem and did not know how to do it beforehand, how would I come up with the solution?
• Why does this seemingly simpler approach not work here? (maybe you are onto something: do not trust the instructor to be infallible)
• You are using words that you have not used before, what does this even mean?
• Can you draw a picture?

As long as you ask because you actually want to learn and are respectful of everyone in attendance, you are in the clear.

You are encouraged to work together on HW and study together. In the few cases when this is not allowed, we will make it explicit. We realize the difficulty of doing so via remote learning. A good place to start is Piazza. Try out asking a question there or attempting to answer someone else's question. Do not worry if your answer might be incomplete or wrong. We will help you out, clarify doubts. If you feel insecure there is always the possibility of posting/answering anonymously on Piazza. But do consider putting your name on your posts: it makes math a bit more human.

Come to office hours! One could say that office hours are really what distinguishes this course from self-learning: you can watch a bunch of YouTube lecture (that may be better than what we teach), but we can actually listen and answer your questions!

Suggested textbooks are listed below.

• [Rudin] (recommended/not required) W. Rudin, 1976. Principles of mathematical analysis, 3rd ed., International series in pure and applied mathematics. McGraw-Hill.
• [Pugh] (optional) C. Pugh, 2017. Real Mathematical Analysis, 2nd ed., Undergraduate Texts in Mathematics. Springer.

Rudin Will be used as a reference text. I find it hard to study independently on Rudin but it works great as a concise source of theorems and exercises.

Pugh Can be used for independently study or to have an alternative approach to the topics covered. It is more verbose and provides gentler explanations.

We will not follow the books closely. I will attempt to provide references for the topics covered in the books but this will not always be possible.

All lectures will be recorded and made available online. Office hours WILL NOT be recorded.

I will present the lecture material through a screen share of a screen that I will use as a virtual whiteboard. All the written material will be made available online. I use OneNote to write notes. I will provide a read-only link to the OneNote notebook that you can open in the browser and that will update in real-time during the lecture. Since the screen real-estate is limited, I suggest you keep it open because it allows you to scroll back and consult what I have written previously if you get lost or confused.

I will provide a detailed per-lecture syllabus in the corresponding section. It will address the topics covered and list the major definitions/lemmata/theorems. I suggest you review the syllabus before and after each lesson to have a preliminary idea of the topic we will cover and as a way to review and make sure you caught everything covered in the lecture.

Preview and review questions are posted there. Please check often.

We will try to make reasonable accommodation for any eventuality that students may encounter. If you have any concerns, do not hesitate to contact the instructor. All lecture material is recorded and available to students. If carrying out the coursework is problematic for whatever reason, we will do our best to be flexible with deadlines on a case by case basis.

This is an upper level math course and it will be challenging: expect to not be able to fully understand or complete some of the work. The instructor is available during office hours both to answer your doubts about the course material and suggest ways to approach studying and coursework.

UVA also provides resources If you are feeling overwhelmed, stressed, or isolated. The Student Health and Wellness Center offers Counseling and Psychological Services (CAPS) for its students; call 434-243-5150 to speak with an on-call counselor and/or schedule an appointment. If you prefer to speak anonymously, you can call Madison House’s HELP Line at any hour of any day: 434-295-TALK. Alternatively, you can call or text the Disaster Distress Helpline (1-800-985-5990, or text TalkWithUs to 66746) to connect with a trained crisis counselor; this is toll free, multilingual, and confidential, available to all residents in the US and its territories.

All students enrolled in Fall 2020 courses, and who have successfully completed a FAFSA for the 2020-2021 academic year, can request funding for expenses related to the disruption of fall campus operations due to the pandemic. For information on CARES Act Student Emergency Funding, Bridge Scholarships, and Emergency Loans, please visit Student Financial Services Operational Updates. You might also be eligible for an Honor Loan.

Attendance is strongly encouraged but not mandatory.

Please do not use mock/joke names in zoom lectures as it can be a disturbance for the class. Go by the name you commonly go by.

Online meetings require some extra attention on behalf of all participants. Please try to make sure your system does not echo and is sufficiently quiet when you are unmuted. We are however understanding of various circumstances that may arise as long as you are acting in good faith. If you are in doubt, please contact us.

When you attend an online meeting, strongly consider turning on your webcam. This provides us with a way to gauge your reaction, understand when we are being confusing etc. Turning on the webcam is a way for YOU to help US do a better job.

The University of Virginia Honor Code applies to this class and is taken seriously. Any honor code violations will be referred to the Honor Committee. Upon submission of each assignments in this class you pledge to abide by the rules of UVA, this course, and the specific assignment. If you have doubts, please ask on Piazza.

You MUST acknowledge any help you received on assignments, even if it is permitted: working with a fellow student, looking up things online, etc. Not acknowledging external help is an Honor violation.

All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered.

• Sets, set operations, Cartesian product.
• Quantifiers
• defn Groups, rings, fields
• \(\N\) and \(\Z\)

• Proofs.
  • English vs mathematical notation.
  • The word "fix" and "let".
  • How to prove equality of two sets/classes of objects.
  • How to prove existence and uniqueness.
• Equivalence relations.
  • defn Equivalence relation on a set \(X\).
  • defn Equivalence class of an element \(x\in X\).
  • defn The quotient set \(\frac{X}{\sim}\)
  • prop Different equivalence classes are pairwise disjoint. proof as an exercise.
• Rational numbers \(\Q\)
  • defn The equivalence relation \(\sim_{\Q}\) on \(\Z\times(\Z\setminus\{0\})\).
  • defn The rational numbers \(\Q\) as a quotient set.
  • defn operations on \(\Q\).
• The issue of well-definedness.
• Order relations.
  • defn non-total and total orders on sets.

• Rudin Chapter 1 pp 1-5
• Rudin Chapter 2 pp 22-35 up to Thm 2.8

• Order relations (non-total and total)
• defn upper/lower bounds.
• defn bounded subsets.
• defn least upper/ greatest lower bound.

• Sequences and subsequences.
  • the words eventually and frequently.
  • the word neighborhood.
  • defn Convergence in \(\Q\).
  • Review: prop the limit, if it exists is unique; prop algebra of limits.
  • the quantifier \(\forall \epsilon\gt 0\) and the possibility of changing \(\epsilon\) with e.g. \(10\epsilon\) in the proof.
• defn Cauchy sequences in \(\Q\).
  • prop All convergent sequences are Cauchy.
  • prop / exmpl \(\Q\) has non-convergent Cauchy sequences.

• Rudin Chapter 1 pp 1-5
• Rudin Chapter 3 pp 47 - 58

• prop / exmpl \(\Q\) has non-convergent Cauchy sequences: the divide and conquer algorithm

• Constructing \(\R\)
  • Cauchy sequences in \(\Q\) \(\mathrm{Cauchy}(\Q)\).
  • Cauchy subsequences with geometric control.
  • The equivalence relation \(\sim_{\R}\).
  • defn \(\R=\mathrm{Cauchy}/\sim_{\R}\)
  • defn Order relation on \(\R\).
  • defn Operations on \(\R\).

• Rudin Chapter 1 pp 1-5
• Rudin Chapter 3 pp 47 - 58

• \(\R\) as a quotient of \(\mathrm{Cauchy}(\Q)\), order, operations.
• Subsequences.

• Rudin Chapter 1 pp 1-5
• Rudin Chapter 3 pp 47 - 58
• Rudin Chapter 2 pp31-32 metric spaces

• Quickly Cauchy sequences.
• Subsequences.

• Completeness of \(\mathbb{R}\)
  • thm \(\R\) is complete.
  • cor \(\R\) has the least upper bound property (on HW01)
• limit points
  • defn \(\limsup\) and \(\liminf\)
  • defn limit points of a sequence
  • rmk techniques of finding \(\lim\) using \(\liminf\) and \(\limsup\).
  • prop \(\limsup\) and \(\liminf\) are, respectively, the largest and least limit points (on HW02).

• Rudin Chapter 1 pp 1-5
• Rudin Chapter 3 pp 47 - 58
• Rudin Chapter 2 pp31-32 metric spaces

• Product metric spaces
  • rmk distances \(d^{1}_{X\times Y}= d_{X}+d_{Y}\) and \(d^{\infty}_{X\times Y}=\max( d_{X};d_{Y})\) are comparable up to a constant \(2\).
  • prop A sequence \(v_{n}=(x_n,y_n)\in X\times Y \) is Cauchy if.f \(x_n\in X\) is Cauchy and \(y_n\in Y\) is Cauchy.
  • prop A sequence \(v_{n}=(x_n,y_n)\in X\times Y \) converges (\(v_{n}=(x_n,y_n)\to \bar{v}=(\bar{x},\bar{y})\)) if.f. \(x_n\to \bar{x}\) and \(y_n\to \bar{y}\).
  • prop \(X\times Y\) is complete if.f \(X\) and \(Y\) are complete
• Banach spaces
  • defn normed vector spaces
  • defn a distance induced by a norm
  • defn Banach space
  • rmk distances induced by norms are translation invariant
  • rmk distances induced by norms are homogeneous of degree \(1\)
  • defn equivalent norms
  • defn the norms \(l^{p}\) on \(\R^{d}\)
  • rmk on \(\R^{d}\) the norms \(l^{p}\) are equivalent for different \(p\in[1,\infty]\)
  • prop all norms on \(\R^{n}\) are equivalent (no proof, yet; we need basic topology)
  • corr All normed finite dimensional real vector spaces are Banach
  • defn The space \(l^{\infty}(\N;\R)\) and the norm \(\|\cdot\|_{l^{\infty}}\)
  • prop The space \(l^{\infty}(\N;\R)\) is a Banach space (left for HW03)
  • defn The space \(l^{\infty}_{c}(\N;\R)\)
  • prop The space \(l^{\infty}_{c}(\N;\R)\) is not a Banach space (left for HW03)
  • rmk How to show that a space is NOT complete: it is better to show that there is a limit in an ambient space that "escapes" from the original space, and then use uniqueness of limits.

• General topology
  • defn open, closed
  • exmpl examples of closed an open sets
  • rmk being open and closed depends on the containing metric space and on the subset
  • defn induced metric of a subset

• Rudin Chapter 2 pp 32-45

• Rudin Chapter 2 pp 32-45
• Rudin Chapter 2 pp 36-41

• Continuous functions
  • defn a function continuous at a point
  • defn a continuous function between metric spaces
  • prop equivalence of metric and sequential characterizations of continuous functions.
• Compact sets
  • proof Heine Borel thm.
  • proof Compact sets are bounded
  • proof Compact sets are complete
  • proof Compact subsets are closed
  • proof Closed subsets of a compact set are compact
• Examples

• Rudin Chapter 4 pp 89-93

• Compact sets and continuous functions
  • proof Image of a compact set is compact
  • proof If \(f\colon K\to \R\) is continuous and \(K\) is compact then \(f\) attains max and min.
• proof all norms are equivalent on \(\R^d\)
• defn \(r\)-nets and defn total boundedness

• Rudin Chapter 4 pp 89-93

• proof a space is compact if.f. it is complete and totally bounded
• rmk Checking total boundedness of a subset \(K\subset X\): no need to make sure that nets are centered in points of \(K\)
• Uniform continuity
  • proof a continuous function on a compact set is uniformly continuous
  • exmpl a non-uniformly continuous function
  • exmpl a uniformly continuous function
  • exmpl a non-uniformly continuous (and unbounded) function on \(\R\): bumps going to \(\infty\)
  • exmpl a non-uniformly continuous (and unbounded) function on \((0,1]\): bumps going to \(0\)
  • rmk Failure of infinite dimensional spaces to be locally compact (closed bounded sets of infinite dimensional spaces are not compact)
  • exmpl generalizing the above example to function on \(\overline{B_2(0)} \subset l^\infty(\N;\R)\) (tips for HW04 exercise)

• Uniform continuity
  • examples
  • exmpl manually checking that a sequence steeper of bumps going to \(\infty\) is not uniformly continuous
  • prop a continuous function on \(\mathbb{R}\) decaying at \(\infty\) is uniformly continuous.
  • rmk boundedness of derivatives and uniform continuity: the former implies the latter but not vice versa
• Extension of continuous functions
  • exmpl a continuous but not uniformly continuous function that has no extension to the closure: \(1/x\) on \((0;1]\).
  • thm a uniformly continuous function \(f\colon E\subset X\to Y\) admits a unique continuous extension to \(\bar{E}\) and that extension is uniformly continuous
  • proof uniqueness of the extension (using only continuity, not uniform continuity)
  • idea how to set up the existence: role of uniform continuity

• Extension of continuous functions
  • proof existence of the extension
    • rmk well-definedness of the extension
  • proof continuity of the extension
  • proof uniform continuity of the extension
• The modulus of continuity \(\omega\)
  • rmk graphical interpertation
• Continuous functions as a Banach space: uniform convergence.

• prop \(x\mapsto \exp(x)\) is continouos on \(\R\).
• exmpl Continuous and uniformly continuous functions
• exmpl Showing or disproving uniform convergence
  • \(n\mapsto x^n\) is not uniformly convergent on \([0,1]\)
  • \(n\mapsto x^n\) is not uniformly convergent on \([0,1)\)
• rmk the uniform limit has to be the pointwise limit
• Equi-uniform continuity and compactness in \(C^0\)
  • defn equi-uniform continuity, equi-uniformly continuous families of functions
  • thm Compact subsets of \(C^0(X;\R)\) with \(X=[0,1]\) are bounded, closed, and equi-uniformly continuous
  • proof Compact subsets of \(C^0(X;\R)\) with \(X\) cpt are bounded, closed, and equi-uniformly continuous
  • thm (sketch) Subsets of \(\mathfrak{F}\subset C^0(X;\R)\) with \(X=[0,1]\) that are bounded, closed, and equi-uniformly continuous are compact
  • thm The above two implications together for an arbitrary cpt metric space \(X\) are called the Ascoli-Arzelà theorem

Assigned: <2020-08-24 Mon> Due: <2020-09-02 Wed>

Assigned: <2020-09-01 Tue> Due: <2020-09-11 Fri>

Assigned: <2020-09-09 Wed> Due: <2020-09-22 Tue>

Assigned: <2020-09-15 Tue> Due: <2020-09-29 Tue>

Assigned: <2020-09-28 Mon> Due: <2020-10-06 Tue>

Assigned: <2020-10-14 Wed> Due: <2020-10-27 Tue>

Assigned: <2020-10-27 Tue> Due: <2020-11-10 Tue>

Assigned: <2020-11-04 Wed> Due: <2020-11-17 Tue>

Assigned: <2020-11-28 Sat> Due: <2020-12-07 Mon>

Assigned: <2020-11-28 Sat> Due: <2020-12-10 Thu>

• Take home assignment.

• Both course textbooks but not any other books, any statement you find in the book in the chapters we covered can be referenced.
• Presonal course notes.
• Class OneNote notebook / recordings.
• Posted HW solutions.
• No external help of any kind.
• No collaboration - strictly individual.

Please make a private piazza post tagged final and with topic: $YOURNAME final topic choosing the final topic (see below). I will suggest specific theorems from that topic I would need to see proven completely, I will also suggest statements that I am happy for you to skip.

Oral exams will be carried out in the following time slots:

• Dec 8 from 8:00 to 21:00
• Dec 9 from 13:30 to 21:00
• Dec 10 from 18:30-21:30
• Dec 11 from 8:00 - 22:30
• Dec 12 from 8:00 - 22:30
• Dec 13 from 8:00 - 22:30

Please suggest a time you would like to do your exam. You should request times that are in 1:15 min increments from the start time of any given time slot. After your presentation, during the presentation of your colleague after you, you will have time to prepare the HW answer in a breakout room.

The timetable is somewhat subject to change. In a given day you will be heard in order. However the time you are heard might end up being later than scheduled. I appreciate your understanding. If you have time constraints please make sure to schedule a time when you are not at risk in case the schedule undergoes modification.

The general topics are listed below. Your presentation should consist of:

• an overview of the topic
• stating important definitions, theorems, and lemmata (proof not necessary or general idea/picture proof)
• proving in detail an important theorem (to be established with me prior to your exam)

Presentations should NOT be slide-based but rather should be done at the moment using Zoom and with one of the following options:

• tablet screenshare
• white/blackboard with camera pointed at it
• overhead camera (cellphone?) pointed at a piece of paper on which you write

I suggest you try out your technical setup before the final. You can arrange for a short meeting with

The general setup of your presentation should be similar to the class lectures with slightly less emphasis on motivation (still should be present) and more emphasis on formal correctness/completeness. You may need to fill in some of the details I ommitted in class.

You will be given the statement of one/two of the problems of which the solutions have been published. You will have 15 min to prepare your answer by yourself (breakout room). This part is CLOSED BOOK. You may not consult ANY notes, books, recordings. You will then present to me the solution of the problems you were asked to prepare. This part should take about 20 min.

The subpoints of the topics listed below do not have to be covered in entirety. I will suggest what you should speak about. I will honor your preference in terms of the large-scale topics. You may express preferences on what you want to focus on among the subpoints but I might suggest modifications to your plan.