This course should be better called Metric Spaces and Some Point-Set Topology;
We have adopted the textbook A.W. Knapp: Basic Real Analysis, Birkhauser Verlag 2005, Chapter II and some parts of Chapter X.
This is one more
opportunity to lay the foundation of a rigorous approach to analysis. In the
calculus and advanced calculus courses, students got the basics of the analysis.
In this course, we go to a more abstract level of topological spaces,with a
special emphasize on metric spaces. However, we will first present many concrete
examples to illustrate the diversity of the concept of a metric space and some
of the complications that may occur.
Analysis is that
domain of mathematics that combines algebraic, order and
topological structures.
The analytical ideas are present almost in all other domains of mathematics and
the aim of this course is to give the basic results in point-set topology and
metric spaces that any mathematician will need under most of the
circumstances.
Metric Spaces and Point-Set Topology (Fall Semester) The actual syllabus
Metric spaces and examples:: definition, Euclidean spaces, Frechet space, discrete metric, post office metric, hedgehog metric, Hilbert cube metric, normed spaces, l_p spaces, normed spaces, inner product spaces.
Topology of metric spaces: accumulation points, adherent points, open sets, closed sets, and their properties, topology on a metric space.
Convergence and continuous functions: characterizations of continuous functions, convergence of sequences, product of metric spaces.
Separation properties of metric spaces: Hausdorff property, regularity, normality, Urysohn property, Tietze Extension Theorem.
Bases of topologies: bases and sub-bases, dense sets, separability.
Compact metric spaces: open coverings, finite intersection property, compactness, sequential compactness, completeness, totally boundedness.
Connectedness: connected spaces, pathwise connected spaces, properties.
Baire Category Theorem: rare sets, meager sets, Cantor type sets, the weak and strong forms of Baire Category Theorem.
Equicontinuity:: equicontinuous families of functions, uniform equicontinuity, Ascoli-Arzela Theorem, applications.
Fixed Point Theorem: fixed points, contractions, Lipschitz functions, applications.
Approximation properties of C(S): Weierstrass Theorem of Approximation, the algebra C(S), Stone-Weierstrass Theorem.
Abstract completions of metric spaces: complete and noncomplete metric spaces, completion.
Topological spaces: definition and basic properties.
It is extremely important to perform
all the homework assignments (normally, five questions). Additionally, we
recommend to all students (at least try) to solve all the other questions in the
textbook. There is no other way of understanding mathematics out of doing
it.
In order to fully understand mathematics, doing exercises is not
sufficient, like in the high school. You should understand definitions,
theorems, examples and counterexamples, and at the last stage of education,
the proofs of the results. This does not mean simply memorizing the
definitions, the statements, or the proofs.
Definitions: 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. Put these definitions and examples in perspective,
in connection with other objects and results; try to ask yourself why 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 collection of properties B is true. In
order to understand a result, it is important to clearly separate the
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).
Proofs: 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. All the implications in a proof
should be checked; if at least one is not correct, the entire construction will
collapse (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 understanding of 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) during the lectures and you
should learn how to do it by yourself later.
Calculations: 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 are met. If not, your
calculations may be formally correct but the final result may be wrong, or at
least not sufficiently explained, 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 become 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%):
|
Points |
Letter-grade |
|
85-100 |
A |
|
80-84.9 |
A- |
|
75-79.9 |
B+ |
|
70-74.9 |
B |
|
65-69.9 |
B- |
|
60-64.9 |
C+ |
|
55-59.9 |
C |
|
50-54.9 |
C- |
|
45-49.9 |
D+ |
|
40-44.9 |
D |
|
0-39.9 |
F |
There will be two midterms and a final written examination, acknowledged in advance.
Rules of this
course:
Attendance will be taken for each hour, separately, of the class. There will be four hours weekly, with the possibility that some of the classes can be canceled. The fourth hour will be usually dedicated to discussions of the homework solutions.
Quizzes. There will be five quizzes randomly distributed. No make up for quizzes is possible.
Homework. There will be about eight homework assignments. Each assignment will consist of about five questions. No make up for homework is possible. Plagiarism will be severely punished.
Basic Rules: Please do not enter later than the first five minutes from the beginning of the class, do not go out of the class except in the real emergency cases, and do not chat during the class. Violations of these rules of behavior will be punished.
Page maintained by Aurelian Gheondea
Last update: 26th of September 2011