This course should be better called Introduction to Analysis;
We have adopted the textbook An
Introduction to Analysis, 3rd Edition, by W.P. Wade,
A few words about Analysis:
This is the first
opportunity to lay the foundation of a rigorous approach to analysis.
In the calculus course,
students have been trained mainly in how to handle derivatives and
integrals, but analysis is much more than this. We start with four
postulates, plus the basic objects and axioms from the set theory. From
here we develop an axiomatic system
and prove rigorously all the
theorems that make this system.
about combined structures: algebraic,
order and topological.
The analytical ideas are present almost
other domains of mathematics and the aim of this course is to give the
basic results and mathematical objects of analysis that any
mathematician will need under most of the circumstances.
It is extremely important to perform weekly the homework
(normally, five questions from the textbook). Additionally, we
recommend to all students (at least try) to solve all the other
questions in the textbook. There is no other way to understand
mathematics except doing
In order to fully understand mathematics, doing exercises is not
sufficient, like in the high school. You shuold understand definitions, theorems, examples and
and at the last stage of education, the proofs of the results.
does not mean simply memorizing the definitions, the statements and the
Analysis (as part
of mathematics) uses mathematical objects that have to be precisely
defined. Understanding the definitions of these objects is the first
essential step in this enterprise. Do not stop after being able to
repeat (memorize) a definition: try
to fully understand the ideas
behind and have sufficiently many relevant examples.
defintions and examples in perspective, in connection with other
objects and results; try to ask yourself why are are they needed (if
they really are!).
Statements of the results (theorems,
lemmas, propositions, remarks):
A statement (the bricks of a
mathematical theory) is generally in the form of a logical implication:
If a collection of properties A holds then some other colllection
of properties B is true. In order to understand a result, it is
important to clearly separate
hypotheses (the collection of
properties A) from the conclusions (the collection of properties B).
Many times, there are some subtle relations between the hypotheses
themselves, and/or between the hypotheses and conclusions. Always ask
yourself why the hypotheses are needed
(maybe they are not needed, or
at least not all of them!). At the end of the day, you should be able
to reproduce the statement from your memory, not necessarily exactly as
in the textbook, but the ideas should be the same. A good exercise is
to try to change the notations (use different letters).
The ultimate goal of mathematics is to provide correct
proofs for the statements. This is the most difficult part and it takes
time and effort to get that skill. Mathematics is different from all
the other sciences in the sense that it is a deductive and formal
system. Roughly speaking, this means that a statement will be accepted
as a result only after at least one convincing logically correct proof
is obtained . A proof is generally a sequence of logical implications.
the implications in a proof should be checked; if at least one is not
correct, the entire construction will colapse (a chain is as strong as
its weakest link). You have to be
convinced by the correctness of each step
in a proof, with no exception
. You should have a clear
all the steps in a proof, why are they needed (maybe some of them are
not, and the proof can be simplified) and what are the relations
between these steps. I am trying to clearly depict the steps in a proof
(especially for complicated and longer proofs) durng the lectures and
you should learn how to do it by yourself later.
The ability of
doing calculations in analysis is taught in Calculus (first year
course). Do not lose this ability
and even more, try to improve it. But
there is something more required at the present level:
calculations hold under certain hypotheses (assumptions) and each time you
want to do a calculation ask yourself whether the necessary assumptions
. If not, your calculations may be formally correct but
final result may be wrong, or at least not sufficiently argumented
which means that you cannot rely on that final result. In doing correct
calculations you prove that you fully understood the results: if we
compare a calculation with a handwork, then the results that you know
are your precious tools that you have to know how to correctly use in
order to get your job well done.
We use the following scale for calculating the
letter grades with respect to the overall calculated grade (1st Midterm
20%, 2nd Midterm 20%, Final 30%, Homework 20%, Quiz/Attendance 10%):
There will be two midterms and a final written examination,
What is covered in this
I. ONE-DIMENSIONAL THEORY
II. MULTI-DIMENSIONAL THEORY
(Spring Semester) Syllabus
- The real number system:
the four postulates, functions, countability, and the algebra of sets;
- Sequences in R: limits
of sequences, limit theorem, the Bolzano-Weierstrass Theorem,
fundamental sequences, limits supremum and limits infimum;
- Limits of functions:
two-sided limits, one-sided limits, continuity, and uniform continuity;
- Differentiability: derivatives,
differentiability theorems, mean value theorems, monotone functions and
the derivative of an inverse function;
- Integrability: the
Riemann integral, Riemann sums, the Fundamental Theorem of Calculus,
improper integrals, functions of bounded variation, and convex
- Series of numbers:
convergence, series with nonnegative terms, absolute convergence,
alternating series, estimation of series, tests for convergence;
- Series of functions: uniform
convergence, power series, Taylor expansions, analytic functions;
- Fourier series:
summability of Fourier series, growth of
Fourier coefficients, convergence of Fourier series.
- Euclidean spaces: algebraic
structure, subspaces and linear transformations, topology ( interior,
closure, boundary, connectedness, and path connectedness).
- Convergence in Rn :
limits of sequences, limits of functions, continuous functions, compact
sets, Dini's Theorem, Lebesgue characterization of Riemann
integrability, Closed Graph Theorem.
- Differentiability in Rn
: partial derivatives and partial integrals,
differentiability, derivatives and tangent planes, the Chain Rule, the
Value Theorem, the Taylor's Formula, the Inverse Function Theorem.
- Integration in Rn : Jordan regions,
Riemann integration on Jordan regions, iterated integrals, change of
variables, partitions of unity, the Gamma function.
- Vector Calculus: Curves,
oriented curves, surfaces, oriented
surfaces, Stokes type theorems.