Isaac Newton

Augustin-Louis Cauchy

Karl Weierstrass

Georg Friedrich Bernhard Riemann

Images from Keith Lynn's "Pictures of Mathematicians" webpage and the The MacTutor History of Mathematics archive.

COURSE: MATH 4217/5217 Call # 33682/33683

TIME AND PLACE: 9:45-11:05 TR in Room 477 of Brown Hall

INSTRUCTOR: Dr. Robert Gardner OFFICE HOURS: TBA

OFFICE: Room 308F of Gilbreath Hall

PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349

E-MAIL: gardnerr@etsu.edu
WEBPAGE: See my webpage ( www.etsu.edu/math/gardner/gardner.htm) for an online copy of this syllabus with homework assignments and any changes which might arise.

TEXT: An Introduction to Analysis, 2nd edition, by J. R. Kirkwood, Published by PWS Publishing Company and Waveland Press, Inc. 1995.

PREREQUISITES: It is assumed that each student has some experience with proof proving (at the level of MATH 2800 - Math Reasoning, for example). Of course, you should feel comfortable with references to results from freshman calculus.

ABOUT THE COURSE: In this course, we give a rigorous development of calculus and a study of the topology of the real line. Several of the results which we will see will be familiar from your freshman calculus classes (in fact, a calculus book will make good supplementary reading). I will occasionally assign problems and cover material not in the text. I will rely on the following sources:

1. Topology, a First Course, by J. R. Munkres. A readable introduction to general topology. This text has been used in the past in our graduate Topology class (MATH 5350).
2. Real Analysis, by H. L. Royden. This is a standard text for a first graduate course in real analysis. It includes the more advanced topics of measure theory, Lebesgue integration and L^p spaces.

GRADING: Homework (H) will be assigned and collected regularly. We will have a midterm (M) and a final (F). Your average will be computed as follows:

The tests will cover:

• Chapter 1 = Sets and Functions, Real Numbers as a Field, Completeness Axiom.
• Chapter 2 = Sequences of Real Numbers, Subsequences, Bolzano-Weierstrass Theorem.
• Chapter 3 = Topology of Real Numbers.
• Chapter 4 = Limits and Continuity, Monotone and Inverse Functions.
• Chapter 5 = Differentiation, Mean Value Theorems.

FINAL: There will be a comprehensive final on Thursday December 15 from 8:00 a.m. to 10:00 a.m. (After trying to find a more suitable time, unfortunately, we will have the final at this originally scheduled time!)

"The Continuum Hypothesis, Part I" by W. Hugh Woodin, in Notices of the AMS vilume 48, number 6, June/July 2001, 567-576. (The second part of the article appears in the following issue of the Notices). This article gives a bit of history of the Continuum Hypothesis and discussion of its relationship to various systems of axioms of set theory.

Everything and More: A Compact History of Infinity (Great Discoveries) by David F. Wallace. W. W.Norton & Company, October 2003. Although the primary aim of the book is to address the work of Cantor, there is also a lot of information on the history of analysis (and even a proof that a countable set has measure 0).

• Problem 1. Find sets A and V such that V is both open and closed relative to A, but V is neither A nor the empty set.

• G-1. The symmetric difference between sets A and B is defined as: A Δ B = {x | x ∈ A or x not in A ∩ B }.
  
  (a) Prove A Δ B = (A \ B) ∪ (B \ A).
  (b) Prove A Δ B = { } (the empty set) if and only if A = B.
  (c) Prove For any set E, (A Δ B) ∩ E = (A ∩ E) Δ (B ∩ E).
• G-2. A number is algebraic if it is the zero of P(x) where P(x) is a polynomial and the coefficients of P(x) are rational. Prove that the set of algebraic numbers is countable. HINT: Show that the collection of polynomials with rational coefficients is countable (this is 1.3.14).

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