Intro To Modern Analysis 1 - MATH S4061 - Summer 2011
Section 2 TUESDAY,
THURSDAY 6:15PM - 7:50PM
MATHEMATICS
407
Instructor: Fabio
Nironi
Email: nironi@math.columbia.edu
Office: Mathematics 415
Tel: (212) 854 4354
Office hours:
Monday from 11:00AM to 12:00AM, Thursday from 5:00pm to 6:00pm;
and
by appointment
TA:
Chris
Hall
Email: chall@math.columbia.edu
Textbook and Readings:
Walter Rudin, Principles of Mathematical Analysis, 3rd Edition (Baby
Rudin!)
This book is renowned for its clarity, synthesis and
rigor. Generations of mathematicians have learned the rudiments of the
art from this book and I will be glad to follow it very faithfully; I
will just occasionally provide some additional reading to complement
it.
Overview of the course:
This is a first course in analysis. We will cover the same topics as
calculus 1/2 plus some additional material (complex numbers, sequences
and series of functions...) minus certain topics that shouldn't belong
in a course in calculus/analysis. The difference between calculus and
analysis is not quite in the contents but rather in the method;
calculus is just about techniques of calculation and doesn't deal with
the deep reasons that support them, analysis on the other hand builds
these
techniques starting from the very foundations of mathematics: logic and
set
theory. A textbook in calculus looks a lot like a cookbook for
men whose wives don't cook, while a book in analysis teaches you how to
cook!
Throughout the course we will deal with many "subtleties" that are
completely ignored in a calculus textbook like the definitions of the
objects that we use; we will try to address certain questions
that you might have considered trivial so far like: can we
switch a limit
and an integral, a limit and an infinite summation or a derivative and
an integral. We will find out that the answer is not always yes and
finding it is never trivial.
We will be working with the complex numbers from the very beginning
even if
this is not a course in complex analysis; the reason is that many of
the results that we will study, work also for the complex numbers just
"out of the box". However, the notion of a "differentiable" complex
function has very little to do with a real differentiable function and
because of this, complex analysis has very little to do with real
analysis.
Grading:
Homework 30%; Midterm 30%; Final 40%
Depending on the circumstances I might decide to evaluate extracredit
assignments.
I pride myself of assigning many A+'s each semester (students must have
at least an A+ on a midterm or the final). The A+ grade is at my
personal discretion and I assign it to people who have an impressive
average (compared to the curve of the class) or who have shown
outstanding improvements through the course.
Midterms:
There will be one midterm exam during class. Make-up exams will not be
given unless a written excuse for missing the exam is provided from
either a doctor in the case of illness or from a dean in other
exceptional circumstances.
Midterm: 7th of July
Final:
The final exam will be on
August
11-th. All students must
take the final at the time scheduled by the
university.
Homework:
There will be weekly written assignments which can be found below along
with the due date. Problem sets are due
on Tuesday in class or
they
can
be
dropped
in
my
drop-box.
The
solutions
will
be
posted on
Courseworks.
- Late homework will not be accepted.
- The two lowest homework grades will be dropped.
- Please staple or paper clip your work.
- Don't forget to write your name on it!
Official Prerequisites:
Multivariable calculus, Linear Algebra.
Real Prerequisites:
None, maybe some familiarity with logic and numbers. This course will
provide the prerequisites to multivariable calculus and partially to
linear algebra; and I strongly suspect that the official prerequisites
have been decided by admin people and not by working mathematicians.
Attending classes:
Attending is not mandatory but is strongly recommended. The book
(unlike the calculus textbook) is really clear and exhaustive, however
because of its clarity and simplicity it might look cryptic and
inaccessible to a novice. If you can read and understand the book like
it was any piece of literature you can safely not come to class.
Help room:
Mathematics 406. There is more information
here.
Calculators:
Calculators ? I really don't see how they can be of any help. However I
strongly encourage you to use a computer to do some independent
research and even experiments. The computer however, is not allowed
during the exam.
Honesty:
Copying your written work from somebody else or from any other source
is considered cheating and will be dealt with severely. Any cheating
during midterms or finals will result in you failing the course and the
matter being reported to your dean.
Additional Material:
Relations and the
construction of numbers
Completion of a
metric space
Detailed
syllabus (part 1)
Stolz-Cesaro
Syllabus (this is just a tentative syllabus and it will be
certainly subject to changes)
The syllabus contains core topics that we will certainly
cover, and a pool of additional topics that we might or might not
cover depending on the pace that we will be able to keep. I will try my
best not to rush.
| Date |
Reading |
Homework |
May 24, 26, 31
|
Naive set theory
Numbers: natural, integer, rational, real, complex
|
Chapter 1: 2
(10pt),
5 (10pt), 6 (25pt), 7 (25pt), 11 (5pt), 12 (10pt), 13 (10pt)
Due Tuesday 7th in class
SOLUTIONS
|
June 2, 7, 9
|
Metric spaces and elementary
topology: open and closed subsets,
compact subsets, Heine-Borel theorem
|
Chapter 2: 5, 6, 7,
8,
10, 11, 12, 14, 16. (5,8 are 5 points, everything else 10 points)
Due Tuesday 14th
SOLUTIONS
|
| June 9, 14, 16, 21, 23, 28 |
Sequences and Series
|
Chapter 2: 13, 15
Chapter 3: 1, 2, 3, 4, 5
Due Tuesday 21st
SOLUTIONS
|
June 30,
July 5
|
Continuous functions between metric spaces
|
Chapter 3: 6, 7, 8, 16, 17, 20, 21,
23
Extra credit: 24
SOLUTIONS
|
July 7
|
Midterm
|
Chapter
1: 8, 9
Chapter 2: 9
Chapter 3: 9, 11, 12, 14,
22
|
|
Differentiation and Taylor polynomials
|
Chapter 4: 1, 3, 4,
6, 7, 13, 17 |
|
Riemann Integral
|
|
|
Sequences and Series of functions.
Uniform convergence, Stone-Weierstrass.
|
|
|
Power series, Taylor series. |
|
....
|
Additional topics: Stolz-Cesaro, Landau symbols and
the principal part of a function, Laplace method, Fourier series,
Algebraic completeness of the complex numbers, Analytic functions
|
|