Course overview

Instructor Contact
Gennady Uraltsev 📧1🌐
Time Location
<Mon 15:30><Mon 16:45> Zoom
<Wed 15:30><Wed 16:45>  
(Recitation) <Fri 15:00><Fri 16:30>2 Zoom

Office hours

Time Location Notes
<Tue 16:40><Tue 17:15> Zoom By appointment on Piazza
<Tue 18:45><Tue 19:20>    
<Thu 18:30><Thu 19:30> Zoom  
  1. Meetings WILL NOT be recorded.
  2. A OneNote link that allows editing will be provided during Office Hours. If you want to be able to write make sure you have a (free) One-Note account and have downloaded the app or can access One-Note online through a browser. The One-Note notebook will be cleared by the end of the day and the link will be disabled.
  3. You can share your screen with Zoom to show me your work.

Summary

The main goal of the course is to develop a working knowledge of basic mathematical analysis in the setting of metric spaces. This course is based entirely on proofs. The course will have some overlap with MATH3310; however we focus on developing the theory in significant generality. Knowledge of basic real analysis as it provides a collection of examples without which the generality of the MATH4310 can be confusing. Throughout the course we also will cover abstract results and specific that go beyond basic real analysis (e.g. Banach spaces as examples of metric vector spaces).

By the end students are expected to have a deep understanding of the relation between concepts of (metric) closeness, continuity, uniformity, and students will be able to recognize these concepts in novel situations.

This is an online synchronous course, which means we will meet for regular class sessions online via Zoom (Collab “Online Meetings”). Your presence and active participation are strongly suggested and are important for creating the most effective and engaging learning experience. You are not expected to be on Grounds for exams or class meetings.

Main topics

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)

Prerequisites

Basic real analysis or equivalent.

You should have taken MATH 3310 (Basic Real Analysis) or equivalent. The most essential skill from 3310 that is crucial in 4310 is being able to work with the \(\epsilon-\delta\) definition of the limit and related notions like convergence and continuity (just formally understanding the definition of the limit is not enough).

Proof writing

You are expected to have substantial prior exposure to proofs. This course is proof based. You should be completely comfortable with basic proof techniques (e.g. proof by contradiction) and have experience in both reading (and understanding) proofs and writing your own proofs. You will refine your knowledge of proof writing throughout the course, specifically in the context of the topics we cover.

Basic notions about limits

Students are encouraged to already have familiarity, in the context of \(\R\), with:

  • a limit is unique,
  • a limit can exists or can fail to exist,
  • sandwich theorem (aka policeman theorem) (with idea of proof).

Basic notions about the real line

Students are encouraged to already have familiarity with:

  • the least upper / greatest lower bound (\(\inf, \sup\)) of a subset of \(\R\)
  • the intermediate values for continuous functions (statement/idea of proof)
  • a continuous function on a closed and bounded subset of \(\R\) is bounded and attains maximum and minimum.

Basic notions about differentiable functions

Students are encouraged to already have familiarity, in the context of functions on \(\R\), with:

  • the derivative of a function in a point,
  • the derivative function \(f'\) of a functions \(f\colon \R\to \R\),
  • derivatives of polynomials, sine, cosine (no proof),
  • derivatives of sums, product rule, chain rule (no proof),
  • finding maxima and minima using derivatives (with idea of proof),
  • Taylor expansion (heard of it).

Basic notions about the Riemann integrals

Students are encouraged to already have familiarity, in the context of functions on \(\R\), with:

  • the Riemann integral as a limiting procedure for finding the area under a curve,
  • the Fundamental Theorem of Calculus (with idea of reason why it works).

Assignments and grading

Homework

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.

Exams

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

Grading

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    
Assignment Percentage Notes
HW (8.5 assignments) 60 Only best 7 homework grades will be kept
Midterm 1 12  
Midterm 2 12  
Final (oral exam) 16  

Grading scheme (subject to slight modifications)

Letter grade scheme  
A- – A+ 80-100%
B- – B+ 68-80%
C- – C+ 54-68%
Passing grade 49%

How to succeed (one way, there are many!)

Questions

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.

Working together

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.

Office hours

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!

Course materials

Textbooks

Suggested textbooks are listed below.

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.

Recorded lectures

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

Lecture notebook

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.

Syllabus

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.

Policies

The value of this course lies not only in the material being taught but in the approaches and perspectives that students have interacting with topics presented. Factors like, social identities, personal circumstances and disabilities, access to specific environments and infrastructure all influence the experience each one of us has of this course. I am committed to giving value to your ideas and contribution, to listening, and to building an environment that rewards everyone for communicating and defending your ideas. I encourage all participants to contribute in the way they best see fit to this goal.

Both instructors and students have to make their best effort to facilitate learning. I encourage you to attend because it is during courses that we can engage with each other and share insights, ideas, and guidance for the material covered. However I also understand that everyone has their own learning style and pace and if you prefer a more independent approach we do not wish to penalize you for it by requiring attendance.

I acknowledge that many forms of discrimination and racism are deeply seated in our society, the system of higher education, and the history of UVA, and I believe that at our actions can either reinforcing or work against the systemic injustices. I commit to listening and holding myself accountable for implementing practices that combat discrimination, and encourage you to help me do so. Students of all racial, gender, immigration status backgrounds are welcome in this classroom. If your status is impacting your ability to succeed in the course please reach out to me to discuss the situation. I commit to listening, being forthcoming, and finding channels of support in full confidentiality unless required by judicial warrant.

With this in mind please consider the following:

Flexibility

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.

Well-being

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

Attendance is strongly encouraged but not mandatory.

Online etiquette

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.

Honor code

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.

Accommodation

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.

Important dates 3

  • Academic Calendar
  • A&S Calendar
  • First day of class : <2020-08-26 Wed>
  • Last day of class : <2020-11-23 Mon>
  • Last day to add / change credit options : <2020-09-08 Tue>
  • Last day to drop class : <2020-10-13 Tue>

Lectures

Lecture 1 <2020-08-26 Wed>

Introduction and overview of the course

Review:

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

Topics

  • 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.

References

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

Lecture 2 <2020-08-31 Mon>

Questions 02 <2020-08-31 Mon>

Review

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

Topics

  • 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.

References

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

Lecture 3 <2020-09-02 Wed>

Questions 02 <2020-08-31 Mon>

Review

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

Topics

  • 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 \(\R\).

Referecences

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

Recitation 1 <2020-09-04 Fri>

  • The ideas and scetches of proofs for well definedness of the multiplication and division, and a bit about how to generally check if an induced operation is well defined.
  • Different constructions of reals and the conditions to check to ensure that they give equivalent structures.
  • Intermediate value theorem and local connectedness of reals.
  • Infinite decimal expressions for reals, and their connection to Cauchy sequences.

Lecture 4 <2020-09-07 Mon>

Questions 03 <2020-09-03 Thu>

Review

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

Topics

Metric spaces

  • defn Metric space
  • expml \(\Q\) as a metric space
  • expml \(\R\) as a metric space
  • exmpl a weighted finite graph as a metric space.
  • defn limits of sequences in metric spaces
  • defn Cauchy sequences in a metric space
  • defn Complete metric spaces.
  • thm \(\R\) is complete (setup)

Subsequences

  • defn subsequences
  • quickly Cauchy subsequences (geometric control of increments)
  • prop A Cauchy sequence admits a quickly Cauchy subsequence.
  • lem A Cauchy sequence has a limit if.f. at least one of its subsequences has a limit (from HW01)
  • lem If a sequence converges then all of its subsequences converge to the same limit (from HW01).
  • rmk to check completeness, one can restrict to finding a convergent subsequence.
  • rmk to check completeness, one can assume that the Cauchy sequence is quickly Cauchy.

References

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

Lecture 5 <2020-09-09 Wed>

Questions 03 <2020-09-03 Thu>

Review

  • Quickly Cauchy sequences.
  • Subsequences.

Topics

  • 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).

References

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

Recitation 2 <2020-09-11 Fri>

  • Density of \(\Q\) in \(\R\), Limit points, and a brief mention of adherent points
  • Completion of general metric spaces, well definedness of extended metric
  • Brief discussions of HW1 problems 3.3, 4.1, 5.3.

Lecture 6 <2020-09-14 Mon>

Topics

  • 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.

Lecture 7 <2020-09-16 Wed>

Questions 04 <2020-09-15 Tue>

Topics

  • 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

References

  • Rudin Chapter 2 pp 32-45

Lecture 8 <2020-09-21 Mon>

  • General topology
    • defn closure, interior
    • prop equivalent characterizations of the closure and of the interior.
  • Compactness

References

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

Lecture 9 <2020-09-23 Wed>

Questions 05 <2020-09-23 Wed>

Topics

  • 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

References

  • Rudin Chapter 4 pp 89-93

Lecture 10 <2020-09-28 Mon>

Topics

  • 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

References

  • Rudin Chapter 4 pp 89-93

Lecture 11 <2020-09-30 Wed>

Topics

  • 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)

Lecture 12 <2020-10-05 Mon>

Topics

  • 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

Lecture 13 <2020-10-07 Wed>

Topics

  • 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.

Lecture 14 <2020-10-12 Mon>

???

Lecture 15 <2020-10-14 Wed>

Topics

  • 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

Lecture 16 <2020-10-19 Mon>

  • proof 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
  • proof (sketch) How to replicate the proof of Ascoli-Arzelà for \(X=[0,1]\) on an arbitrary compact metric space \(X\):
    • choosing an \(\epsilon\)-net of points
    • writing an interpolating function without a linear structure.

Lecture 17 <2020-10-21 Wed>

  • Differentiation: a motivation
  • The space \(\mc{L}(X;Y) \) and \(\|\cdot\|_{op}\)
    • prop (from HW02) \(\|\cdot\|_{op}\) is a norm on \(\mc{L}(X;Y) \)
    • prop \(\|BA\|_{op}\leq\|B\|_{op}\|A\|_{op}\) for \(A\in\mc{L}(X;Y) \) and \(B\in\mc{L}(Y;Z)\)
    • proof \(\mc{L}(X;Y)\) is Banach if \(Y\) is Banach
  • The differential
    • prop small \(o\) notation
    • prop the differential is unique
    • prop the differential for \(f\colon\R\to \R\): the limit definition
    • prop Optimization and critical points.
    • thm Lagrange's theorem in \(\R\) and for \(f\colon X\to Y\).
    • prop product rule
    • prop chain rule
    • defn the derivative function, the space \(C^1(X,Y)\).
  • The space \(C^1_b(X,Y)\)
    • prop \(C^1_b(X,Y)\) is a Banach space and \(D\colon C^1_b(X,Y)\to C^0_b(X,Y)\) is continuous.

Lecture 18 <2020-10-26 Mon>

Lecture 19 <2020-10-28 Wed>

Lecture 20 <2020-11-02 Mon>

Lecture 21 <2020-11-04 Wed>

Lecture 22 <2020-11-09 Mon>

Lecture 23 <2020-11-11 Wed>

Lecture 24 <2020-11-16 Mon>

Lecture 25 <2020-11-18 Wed>

Lecture 26 <2020-11-23 Mon>

Assignments

Questionnaire

Assigned: <2020-08-24 Mon> Due: <2020-08-25 Tue>

Homework

Homework is going to be assigned at latest by Tuesday evening or Wednesday morning and will appear on this website. Homework assigned on Tuesday or Wednesday may cover the material covered in class that week, including on Wednesday.

Homework is due by the end of Friday of the following week unless otherwise specified. This means you have more than a week to work on any given homework assignment even though there is 1 HW assignment per week.

It is strongly suggest that you take a look at the HW assignment as soon as it is out. It very well may be that during lecture you will receive hits or ideas on how to approach problems. Taking a look in advance will help you know what to be on the lookout for.

Homework should be turned in via GradeScope. Please mark the sections accordingly when uploading your work.

Specify who you worked with or any external sources of information used. Failure to do so is a UVA Honor Pledge violation.

Some exercises may be difficult. Do not get discouraged if you do not manage to solve everything on your own without a hint. Ask on Piazza or during office hours.

You are invited to collaborate on homework with your fellow classmates, but you must each turn in individual work. You are welcome and encouraged to come to office hours and we allow discussing homework. We do expect you to have worked on it on your own before asking for advice (that's why we put office hours close to the due date).

Solutions will be posted after the deadline for submission.

If you believe that there is a typo/mistake/imprecision in the HW please post your concerns on Piazza as soon as possible. If you are not sure, you may post on Piazza anonymously or send a private messages to the instructor.

HW00

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

HW01

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

HW02

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

HW03

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

HW04

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

HW05

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

HW06

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

HW07

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

HW08

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

HW-extra

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

Exams

Midterm1

Assigned: <2020-10-06 Tue 23:59> Due: <2020-10-11 Sun 23:59> (no extensions)

Midterm2

Assigned: <2020-11-18 Wed 23:59> Due: <2020-11-26 Thu 11:00> (no extensions)

See Piazza post for specific instructions on deadlines for midterm02.

No HW is assigned the week of the Midterm.

Topics

Format

  • Take home assignment.

Allowed material:

  • 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.

Final

The final exam is an oral exam. The first 30-40 min consists of a presentation you prepare on one of the topics of the course. The second 20-30 minutes consist of you solving one/two exercises for which solutions have been published.

Logistics

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.

Presentation format

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.

Homework solution problem

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.

Topics for presentations

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.

Topology of continuous functions

  • Continuous bounded functions as a vector space
  • \(\|\cdot\|_{\sup}\) norm and completeness
  • \(\|\cdot\|_{\sup}\) and pointwise convergence
  • uniform continuity
  • compact sets of \(C^0_b\) (Ascoli-Arzelà)
  • closure of \(C^0_c\) in \(C^0_b\) and \(l^\infty_c\) in \(l^\infty\)

Linear bounded operators and differentiation between Banach spaces

  • The space \(\mathrm{Lin}(X;Y)\)
  • Differential of a map \(f\colon X\to Y\)
  • Lagrange theorem for \(f\colon X\to Y\) assuming the one for \(g\colon \R\to \R\)
  • Completeness of \(C^1_b(X;Y)\)
  • Statement of Taylor expansions for \(f\colon X\to Y\) (optional, no proof, example on \(\R^2\))

Higher differentiability

  • Taylor expansions
  • Taylor expansions with quantitative estimates
  • Completeness of \(C^n_b(X;Y)\)

Integration and the Cauchy problem

  • Defining Riemann integrals: partitions, mesh sizes, Riemann sums
  • Fundamental thm of calculus
  • The Cauchy problem

The Riemann Stieltjes integral

  • Definition and construction
  • Continuity
  • Rough integration

Metric spaces ( a subset of the following topics)

  • Existence and uniqueness of the completion
  • Compactness: equivalent characterizations of compactness
  • The distance function
  • Heine Borel
  • Equivalence of norms on \(\R^d\)
  • Uniform continuity.

Footnotes:

1

Students of this course should use Piazza for all course-related communication, e-mails about course topics will be ignored WITHOUT WARNING.

2

Subject to change.

3

These dates are provided for reference only. Please consult the official UVA registrar.