Jumpstart - Analysis for Graduate School
Introduction
The successful graduate student exhibits a pro-active and inquisitive
attitude towards learning and possesses or can develop an ability to
self-assess and identify and solve his or her own weaknesses and
shortcomings in arguments, ideas, and proofs. Some students have these
traits, some can develop them with training and dedication, while, for
others, they remain elusive.
While much of the material of these lectures is
presented at American undergraduate institutions, most students
merely make a superficial acquaintance with the central concepts of
analysis. It is paramount to gain a deeper understanding of them in
order to be well-equipped to sail smoothly through graduate
school. The emphasis is really on developing an ability to do and
solve rather than to know. According to Lloyd Alexander We learn
more by looking for the answer to a question and not finding it than
we do from learning the answer itself. You are invited to figure
things out for yourself: graduate school does not merely consists in
extending your knowledge of facts, but, rather, in a mathematical
growth driven by understanding. The latter can only be achieved by
personally engaging in the thought process. It is not enough to sit at
the receiving end and collect mathematical facts. Like sports or
music, mathematics requires constant mindful practice for skills to
develop. This can only be done if one cares about going to the bottom
of things, about developing one's own understanding, and about
investigating the multifaceted beauty of mathematical objects and
concepts. A graduate degree is not something that can be given to you,
but something you can become if you allow yourself the possibility
of failing and enjoy the pleasure of exploring new territory while
developing the necessary tools to conquer it.
Every lecture of this course opens with a hidden (until you click on
it) motivating paragraph which aims at providing a broader perspective
pointing to a choice of good reasons why a given topic is
important. The main thread of the lectures is quite standard and takes
you through a selection of important concepts and
results with focus on mathematical rigor. Occasionally formal rigor is
temporarily suspended to provide a word of intuition. Mathematics is,
after all, often an exercise in translating raw intuition into
rigorous, solid, and verifiable arguments. In order to gain a really
useful understanding, you will need to develop your own personal
intuition. The fact that proofs are hidden (but become available upon
clicking) is to be considered an invitation to try and give your own
proof before even reading the one provided. Even after reading our
proof, you are encouraged to hide it again and try carrying it out on
your own. In so doing, you will be able to verify whether you merely
followed its logical steps or you really understood its idea, in which
case you would be able to reproduce it. With persistence and time you
will inevitably develop your own proof and mathematical thought
skills. Not only will you be able to reproduce proofs but also propose
your own. As an added benefit, you will understand the boundary of
validity of results as you will be confronted with the task of
identifying necessary and sufficient conditions. Examples play a
crucial role in testing the range of validity of your mathematical
beliefs and in developing an intuition for abstract concepts, which
are often born out of example. One of my teachers invited students
to always try and find the simplest non-trivial example.
Mathematics requires both "computational power" and "abstract
processing, structuring, and pattern recognition prowess". In this sense
the exercises and problems in the lectures are not only "an hour spent
in the gym" but also an invitation to reflect on important concepts,
their limits, and their relationships.
The final goal is not to have mnemonic access to mathematical facts
but rather to make sense of reality by using mathematics, to develop
your own mathematical worldview.
Learning with understanding usually happens in tandem with growth,
which only occurs in response to a mindful effort. While the first step
towards a deep understanding is the ability to reproduce an argument
(beyond mere mnemonic recall), the ultimate goal is the development of
your own argument. Hence the importance of not simply moving on after
reading a proof and following all of its steps, but to verify whether
you are capable of, at first, reproducing it, and, with time, to
simply "see it" and thus even be able to give alternative proofs. It
is often easy (and misleading) to "get the rhyme but not the reason"!
Remember: there are many explanations for just about any interesting
mathematical fact, but the best of them all is always YOURS! Never
call it quits before you find it. Enjoy the journey!
User's Guide
Lectures
The lectures are structured as follows. They always open with a
motivating paragraph which we ecourage you to read if you are
wondering why at all to even consider the topic of the lecture. You
can then quickly read through a lecture to obtain an overview of the
material. On a second reading you are encouraged to attempt solving
the given exercises and examples in preparation for a better
understanding of the results presented. Finally you should try and
give your own proof of the results, read and understand the one(s)
given, and eventually, try to reproduce it(them), or complete your own, if
you were not able to finalize your first attempt. At this point you
are ready to take on the corresponding weekly problem sets. You can
use the submit a question link in the contact menu to send us
questions/comments about the material covered. Please make extensive
use of it as your inquiry will, of course, be addressed, while,
simultaneously, allowing us to improve the notes over time. Answers to
recurring issues will be integrated into the lecture notes.
Assignments
A weekly homework set will be assigned with the given due date. Please
turn in your solutions in latex format by using this template replacing
Fname and Lname with your first and last names and n
with the homework number. You can simply follow the instructions given
in the file itself in the areas commented out. If you don't have a
LaTex distribution on your computer, we suggest that you download and
install some version of it. Recommended choices are
MacTex for Apple computers
running Mac OS X.
MikTex for computers running MS
Windows.
The LaTex Project
for computers running any operating system or if you prefer not to
download LaTex onto your computer and use an online LaTex typesetting
platform such as
Overleaf instead. Notice that UCI has an institutional
subscription to Overleaf and the platform is therefore available to your for free.
Questions and Answers
In order to foster interaction and participation, you are asked to
submit at least one written question each week about the material
covered in the week's lectures and to answer one of the questions
posted by one of your peers. For this purpose we will use a common
Overleaf file, where the questions and answers will appear and
where everybody can leave their comments, share their ideas, and
suggest topics of discussion.
Particularly enlightening questions and answers will eventually be
posted on this website.
Videos
Videos are intended to give you a brief introduction to important
aspects of the topics covered in the lectures. They should not be used
as a replacement of the notes but rather as an additional motivation
to dwell deeper on the topics. Notice that they do not cover all the
material included in the corresponding lecture.
Disclaimer
These notes are the fruit of the combined effort of Patrick
Guidotti and Song-Ying Li. In spite of our attempt to keep the
lectures error and misprint free, there will inevitably be some that we
overlooked. If you identify any, we kindly ask you to let us know by
using the emailing link available in the top menu under contact. This introduction, the motivational paragraphs at the
beginning of the lectures, and the videos reflect the "taste" and
choices of the first author and are solely his responsibility.