Math 131 B – Real Analysis, Spring 2020 UCLA

Midterms: Friday April 24th. Friday May 15th. Take home exams.

Final: Tuesday June 9th

Google calendar signup link: Here

Lectures: Mondays 12pm + recordings.

Wednesdays 11 am (Except first week, 12pm)

For students in different time zones: Wednesday/Friday 3pm China Standard Time, which is 9 am Central European time. Lectures give
an overview of the topics, main statements and main examples. We won’t do all details always, but focus on the bigger picture and
important proofs.

Videos: Lectures will be supplemented by 6-8 weekly ~15 minute videos on concepts, theorems and proofs, that
help you digest the weekly material and focus on important points. These have more detail and replace portions of the lectures.
Questions on these are taken on campuswire, and can be discussed in office hours.

Office Hours/Discussion/Interactive hours: Thursday 10am, Tuesday 11am. Friday 3pm China Standard Time. These will involve previous questions, live questions, discussion on HW, and some interactive
assignments (brainstorming on a problem in a group after a short introduction). Some content may be recorded and available online.

General comment: With the evolving situation of the virus, all of these policies are subject to change and update during the course.
The grading policies, to the extent possible, will be kept fixed. Delivery of lectures may change though.

Overview of the course: This is a continuation of Math 131A. It will allow us to delve deeper into the topics. A general theme
is to take concepts from 131A and take them to higher dimensions. We will start by taking analysis from the real line
to higher dimensions. The appropriate setting is to talk about general metric spaces. This leads to many profound ideas, like
compact sets, function spaces etc. Then, we take our treatment of series and start talking about series of function. As specific
applications we will discuss the most important ones for applications: Fourier series and a bit of Taylor series. At the end
we spend on about a week on higher dimensional derivatives and differentiation.

Why take this course? This course gives you the tools to more fully understand and apply analysis. Indeed, it focuses on the issues
that you actually encounter in practice: series of functions, higher dimensional spaces, and spaces of functions. The most fantastic applications
of real analysis are in this area and the exciting chance that we have is to cover these ideas.

In the process we get a powerful and general language to talk about convergence and phenomena in any type of space. Indeed, we learn precise language for
open, closed and compact sets. We learn to use abstract tools to reason about these concepts. We get tools to study series of functions and begin to understand
the issues that happen at the limit of such functions. Finally, the ideas about convergence and approximation get a culmination in higher dimensional differentiation.

Course Book(s): We will generally follow Terence Tao Analysis II. Available
Here . However, this book does not contain many detailed proofs,
and this is one of the most important skills to learn, so we supplement the book with two sources.

• Rudin, Principles of Real Analysis. Specifically, Chapters 1, 2, 3, 7,9.
• Charles Pugh, Real Mathematical Analysis: See here.

I recommend to read Pugh for a comprehensive introduction, Tao’s book for a nice summary and problems, and Rudin for a real hard core treatment.
Pugh has many beautiful images and more explanation, and so may be useful to many. There will also be some notes and videos available. Tao and Pugh need
not be purchased as they can be accessed via Springer Link, using UCLA VPN. See here: VPN instructions.

Further, you could consult the following.

• For review and basics 3,4, 7 from: K.A. Ross, Elementary Analysis: The Theory of Calculus, 2nd Ed.
  Available Here .
• Abbott, Understanding Analysis.
• Terence Tao, Analysis I Here.
• For more topology, see Munkres, Topology.

For problems, consult say the following problem books in addition to the problems in Tao’s book and Rudin.

• A problem book in Real Analysis (last few chapters): Available Here
• First chapter of this problem book .

Many other references may be found on springer link. The books with links above can be accessed there with a UCLA VPN account set up. See, the UCLA library instructions for this.

Skills to learn and that will be tested:

• Ability to work with metric space concepts and begin to appreciate topological thinking.
• Use metric/topological concepts like open, closed, compact, bounded, boundary, continuity, limit comfortably and understand their relationships.
• Enhance ability to read proofs, statements and understand them.
• Understand difference between pointwise and uniform convergence.
• Define metric space concepts
• Introduced to the idea of function spaces
• Be able to prove and discuss convergence of series and sequences of functions.
• Be able to discuss examples and define multivariable differentiability and difference of directional derivatives and a full derivative.
• Define continuity in metric spaces and ability to define a relevant space of objects.
• Advanced proof writing skills: clarity, preciceness and explaining your reasoning.

Instructor: Sylvester Eriksson-Bique, available via email and zoom..

TAs:

• Zachary Smith SMITH, Tuesdays 12-13. (TBD, but most likely zoom.)

Time zone and technology:

All times of exams, due dates and lectures are Pacific Standard Time. If you are in a time zone significantly different from this (say 3 hours or more), and
it is difficult to fulfil a requirement due to this, then please let me know. HOWEVER, the deadlines will generally
be designed to give a large window (exams say 24 hours, HW available for a full week). So, I will generally just ask you to submit before and start earlier at a time
appropriate for your time zone. Only in exceptional circumstances, with a compelling reason will I consider extensions.

Also, technology can be glitchy, and may result in delays. Further, laptops can break down or you may experience difficulty connecting
due to firewalls or other issues.
Please do not hesitate to contact me and I will work to solve such problems on a case-by-case basis. You can use UCLA VPN see here for instructions how to set up.

Lectures: Lectures will generally be zoom lectures which are recorded. Some lectures may be recorded in advance and shared, and supplemental
shorter videos may be included in the channel. Office hours will be zoom office hours and some parts may be recorded. Understanding difficulty with time zones bandwidth,
lectures and non-interactive components of office hours will be given online. Attendance in lectures is highly recommended.

Ideas to increase usefulness of lectures/office hours:

• Mute yourself and do not start video. Unmute if you want to ask a question.
• Participate in a low distraction environment.

Submitting as PDF: The best outcomes come with using phone or tablet based scanning apps, such as Camscanner: https://www.camscanner.com.
Other options: Tiny Scanner, Genius Scanner, Turbo Scan, Notes on Iphone, Adobe Scan, Microsoft Office Lens, Doc Scan. Any one of these should work. Make sure
to submit as a single PDF onto gradescope directly, and then also lable your pages to match problems. These speed up grading by a ridiculous factor.

Exams: Two midterms, April 24th and Friday May 15th. Final exam June 9th.

All exams take home exams with a minimum 24 hour window to complete. Turned in online as single pdf. No collaboration allowed, and can not
ask for outside help. Do not discuss exam at all during its time. Otherwise allowed to use all resources, but can not copy. Plagiarism and fraud,
besides being against the academic conduct code, may also be illegal.

Accommodations: With the existing accommodations most likely designed for in-class exams, and understanding the difficulty to get new ones on campus, I will
evaluate and welcome ideas on a case-by-case basis. Please submit requests well in advance and explain the issue with the policy. I will also try and assume
flexible policies, with ample time, to allow for individual variation. This way most extra time requests may not be needed.

In general, I realize we live in very different situations, and each country will be experiencing difficulties with different schedules. I want to support you and let
you know, that I understand that it is a hard time. It can also be a very hard time mentally for us, beyond this class. Please talk to people, and consider
counseling as well. If you suffer individual loss, I feel very sorry for you, and hope I can help in some way. It is a time to support each other, to listen and to
stand together in the face of the difficulties we must encounter. In light of all this, at any time, please don’t hesitate to contact me and let me know how I can help you
and help you individually. All of the policies announced here are flexible, and have room for individual and case-by-case considerations, as needed. This doesn’t mean
I will accept all requests, but that I will at least listen and work to understand.

Homework: There will be weekly homework posted on CCLE (excluding first week and some other weeks as announced.).
It is due at the end of the day Friday.

• No late homework will be accepted, unless you request and are approved an extension for a legitimate reason (unlikely scenario, and strictly enforced).
• Usually 4 problems graded, and remaining graded for completeness.
• Write your name, ID number.
• Submit online on CCLE, must be a single pdf.
• The weakest homework score will be omitted.

Discussion sections: Discussion sections will be weekly and announced by TA.

Grading:

• Normal: Homework, 50%; Midterms 15 % each; Final 20%.
• With take home exams, exams can generally not be missed unless a compelling reason exists (e.g. 16 hour flight during the day).
• Final exam can not be missed. If missed, it will lead to an incomplete or F, and require retaking class/final in a different quarter.

Extra credit opportunities: We will use an online forum, and I will reward 0-0.5-1 % extra credit for this, depending on activity. I think individual
study projects are also an excellent way to learn more. I will post a list of topics that you can study deeper and write a short 3-5 page report on a concept
not fully covered in class, but closely related. You may choose said topic, and collaborate on it. It is worth 0-1-2-3 % depending on quality. Each
project should include a main theorem, at least one definition, at least two examples on them, and a proof of the main theorem, or if it is not feasible,
a proof outline, and arguments for auxiliary lemmas. More detail to be added later.

Inclusivity: This course is welcoming to everyone and from any background. If you at any time feel
a policy of this course causes excessive hardship or is difficult to satisfy for whatever reason (e.g. office hours
conflict with work, or exams conflict with religious holiday), please do let me know ahead of time and we will
work to find an accommodation. If you encounter problems in your personal life, or medical issues, please do reach out for help,
and I can also do my best to forward you to university resources that can help you.

CAE brochure

My goal is to foster an inclusive, supportive and encouraging enviroment for you to learn. Many of us suffer from various degrees
of learning difficulties, exam stress etc. Please be aware that the university has many resources to help with challenges you might face
and that I as an instructor will do my best to assist and accomodate different learners. Please do not be afraid to reach out to a councelor,
me or your TA for contact information to various services. Also, please review the attached brochure.

https://www.cae.ucla.edu/learning-disabilities-brochure