** Instructor Of Record: ** Dr. J.D. Mireles James.

** Office: ** SEC 262

** Office Hours**: Monday, Wednesday, and Friday 11am-11:50pm

(or by appointment).

** Contact: ** The best way to reach me is by email at: jmirelesjames@fau.edu

** Class Location: ** Business Bldg, Room 403.
** Class Time: ** MWF: 12:00pm-12:50pm

** Textbook: ** Elias Stein and Rami Shakarchi ``Real Analysis:
measure theory, integration, and Hilbert spaces''.

Professor Tomas Schonbek has taught the introductory analysis many
times at FAU. The following comments and advice are taken from his
course description:
"Introductory Analysis is a two course sequence whose main purpose is to teach
the basics of analysis in a rigorous and reasonably complete way. In our current
setup, Part I can be called Calculus of one variable "done right." Part II covers
the basics of Lebesgue integration. Together with the Introductory Algebra sequence
it is supposed to prepare students for more advanced courses in mathematics. In our
doctoral program, the qualifying exams are designed to test your readiness for taking
advanced courses; an immediate consequence of doing well in the Analysis sequence is
that you will also do well in the analysis qualifier.
Of course, "doing well" does not
mean simply getting a good grade, it means having understood the material, being able
to prove the theorems, and being able to work most of the exercises on one's own. It
may be useful to mention that **writing carefully, expressing yourself in coherent
sentences in the English language (both in writing and orally),
understanding concepts, and proving theorems are essential ingredients of the
course.**

A few historical remarks may be in order. For many centuries, millennia even,
mathematics was a relatively static science. Mathematicians studied properties of
geometric figures, of numbers, and made many marvelous and beautiful discoveries:
The existence of irrational numbers, Eudoxus' Theory of Proportions
(which effectively resolved the crisis caused by the discovery of irrational numbers),
Euclid's proof of the infinitude of the set of primes, Archimedes' sequences to
compute π, his Method in general, Apollonius' work on conics, Diophantus on
Number Theory, the introduction of zero and negative numbers by Indian mathematicians.
The list could go on. But then, beginning in the sixteenth century with a few baby
steps, more in the seventeenth century and finally, coming to fruition with the work of
Newton and Leibniz, a sea change occurred; mathematics became dynamic.
Functions appeared. In his *Principia* Newton studied what he called
*fluents* (functions), and explained how to find the
*fluxion* (derivative) of a fluent and, more interestingly, how to find
the fluent given the fluxion (integration). From then on functions played a central
role in classical mathematics, in analysis as well as in algebra, even though a first
precise definition of the concept did not appear for quite a while. It can probably
be traced to the work of Lejeune Dirichlet in the mid nineteenth century. Ironically
enough, when rigor was taken to extremes in the twentieth century, it was found that
the function, this most dynamical of objects, had to be defined in a static way
as a set of pairs.

Not surprisingly, we will be spending a lot of time studying properties of functions, such as continuity, differentiability, integrability. We will, of course, also spend some time studying the sets on which these functions act, most particularly that extremely complicated and strange set known as the set of real numbers. We have to be rigorous. Newton, Euler, had a tremendous intuition into what they were doing. Most of us are not so fortunate and we need more guidance; rigor and precision provide that guidance. You can think of mathematics as a journey, an expedition, a trek. The journey at times can get quite difficult, but it is also very rewarding. Introductory Analysis and Introductory Algebra are the beginning of the journey, and a lot of time might be spent in doing the analog of fitness exercises, toughening you up for the difficulties that will come. You may find some of these exercises silly and boring, but there still is a point in doing them. You may find some of the going difficult but we assure you that if you stick to it, put a serious effort into mastering the basis."

There will be **no** make-ups. However, there will be ways in which you
can make up for a missed exam or homework, as long as your grades are reasonably good.

---This set of notes states and proves the classification theorem for Riemann integrable functions. (view here).

---This pdf contains the first 20 pages of a book called "Numbers, sets, and axioms: the apparatus of mathematics." (view here).

The discussion of models satisfying the Peano axioms is very complete, i.e. there is a nice uniqueness result. I have not seen the rest of this book but my guess is that it would be very good for learning set theory.

--- A technical, but very well written introductoin to set theory can be found (here).

This is more set theory than we will need this year.

--- A set of notes which discusses the construction of the natural numbers from a set theoretic point of view is (here).

--- A short set of notes discussing the natural numbers from a purly axiomatic point of view is (here).

Note that the axioms are stated slightly differently than the way I did in class, but that they contain the same information.

--- A more complete set of notes discussing the axiomatic view of the natural numbers (is here).

Here the axioms are close to the form I stated in class.

---A note about building the integers out of the natural numbers (view here).

The note goes over the construction we did in class but goes through all the details about extending addition, multiplication, the order relation, and subtraction to the new set.

Warning: the author uses the notation ``omega'' to denote the natural numbers, whereas in class we use a script ``N''.

---A note about building the rational numbers out of the integers (view here).

This note goes over the construction we talked about in class but goes into more detail about general algebraic fields.

---Here is a note about the construction of the real numbers using Cauchy sequences. (view here). The treatment is very thorough, and similar to what I did in class. The only problem with these notes is that the least upper bound property is not established directly. (Though it can be proven from the completeness property, which the author does establish).

---Here is another note about about the construction of the reals. (view here). In these notes you can find the proof the the construction satisfies the least upper bound property.

---Here is a list stating the axioms for the real numbers. (view here).

---Here is a short note which gives a proof that Q does not have the least upper bound property. (view here).

---A short note which shows how to prove that the square root of 2 is a real number. The proof of course uses the least upper bound axiom. (view here).

There will be three in class exams. Each in class exam accounts for 15% of the grade with the final exam accounting for 25% of the grade. Class participation takes care of the last 5% of the grade.

**Course outline** The plan is to cover Chapters 1 (briefly),
2, 3, 4, 5, 6, and 7 of Rudin's book; in other words, the following topics:

- Riemann integration and measure zer. (two weeks)
- Lebesgue measure and Lebesgue measurable functions (two weeks)
- The Lebesgue integral (two weeks)
- Convergence theorems and applications (two weeks)
- Differentiation (two weeks)
- Classical Hilbert spaces (two weeks)

**Grading** Your final grade will depend on homework, a midterm
exam, class participation, and a comprehensive final exam. Homework will be
assigned frequently.
Roughly speaking an A is 90 percent or above, a B is 80-89 percent, a C is 70-79
percent a D is 60-69 percent and below a 60 percent is a failing grade.
However I reserve the right to modify these definitions a little (in your
favor) at the end of the semester.

**Tentative Exam Schedule**
First exam, first week of Feb. Second exam, mid March. Third exam, late April.

Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty is considered a serious breach of these ethical standards, because it interferes with the university mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive of the university community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility. Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4.001.

In compliance with the Americans with Disabilities Act (ADA), students who require special accommodation due to a disability to properly execute coursework must register with the Office for Students with Disabilities (OSD) -- in Boca Raton, SU 133 (561-297-3880) and follow all OSD procedures.

** Course Webpage:**
Check the homepage frequently as it will contain important course updates
including homework assignments.

1) Class has started... welcome!

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