MAT 320: Introduction to Analysis

Stony Brook University - Fall 2016

A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series. This course and MAT 319 meet together for the first 6 weeks of the semester. After the first midterm, the classes will split: students whose grade on the first midterm was higher than a certain cutoff can move to MAT 320 (or amy choose to stay in MAT319). Students whose grade was lower then the cutoff will stay in MAT 319.

Prerequisites

• C or higher in MAT 200 or permission of instructor;
• C or higher in one of the following: MAT 203, 205, 211, 307, AMS 261, or A- or higher in MAT 127, 132, 142, or AMS 161

Lectures time and location

Tuesdays and Thursdays 10:00am - 11:20am in Light Engineering 152,

Lecturer

Oleg Viro
Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University
Arrived at Stony Brook in 2007.

Office: Math Tower 5-110
Phone: (631) 632-8286
Email: oleg.viro AT math.stonybrook.edu
Web page: www.math.stonybrook.edu/~oleg

Research fields: Topology and Geometry,
especially low-dimensional topology and real algebraic geometry.

Office hours

Tuesdays and Thursdays 7:00pm - 8:00pm in Math Tower 5-110.

Tuesdays 11:30am - 12:30pm in Math Tower P-143.

Teaching Assistant

Jean-Francois Arbour, PhD Student
Arrived at Stony Brook in 2015
Office: Math Tower 2-122
Email: jean-francois.arbour AT stonybrook.edu

Homeworks

Homework sets will be typically assigned weekly and due on Thursdays in class. They will be posted on this web site. (NOT on blackboard site!) Late homework will not be accepted. However, grades for homework assignments may be dropped in cases of documented medical problems or similar difficulties.

Exams

There will be two in-class midterms; the first midterm will be on Th, 9/29. The date of the second midterm will be announced later.

Grades will be based on the following scheme: Homework -- 20%; Midterms -- 20% each; Final Exam 40%.

Textbook

Elementary Analysis: The Theory of Calculus, by Kenneth Ross, 2nd edition

Program of the course

Introduction: what are real numbers? Integer numbers and induction; rational numbers and idea of a field; basic properties of real numbers. The completeness axiom. Archimedean property. Infinity, boundedness and unboundedness.

Sequences. limit of a sequence. Limit laws for sequences and formal proofs. Monotone and Cauchy sequences. Subsequences. Bolzano-Weierstrass Theorem, lim inf and lim sup

Topological and metric spaces. Axioms in terms of neighborhoods and open sets and their equivalence. Interior, exterior and boundary points of a set in a topological space. Metric and metric spaces. Balls and spheres in a metric space. Metric topology. Topology of a subspace.

Continuous maps. Definition of continuous maps between topological spaces and their simplest properties. Continuity at a point and its relation to continuity. Sequential continuity and its relation to Continuity. Theorems about operations with continuous functions. Theorem on sequential continuity of composition of sequentually continuous maps.

Extreme Value Theorem. Sequential compactness. Sequentially compact sets in a metric space are bounded and closed. The converse statement for subsets of Euclidean space. Continuous image of a sequentially compact space is sequentually compact.

Intermediate Value Theorem Connected spaces. Properties of connected sets. Connected components of a topological space. Theorem about continuous image of a connected space. Theorems about continuity of monotone functions.

Uniform continuity. Uniform continuity of a continuous function on a closed interval.

Series Convergence and divergence of series, the sum. Geometric series. Cauchy criterion for convergence of series. Tests for convergence of series. Comparison test. Absolute convergence. Ratio and root tests. Harmonic series and its divergence. Convergence of $\sum\frac1{n^p}$. Integral tests. Alternating series theorem. The Riemann rearrangement theorem about conditionally convergent series. Invariance of the sum of a positive convergent series under permutations of its terms.

Power series. Convergence radius of a power series Uniform convergence of a sequence of functions. Continuity of the uniform limit of a sequence of continuous functions. Continuity of power series. Theorem about limit of integrals and integral of the limit

Derivatives Definitions of derivative. Continuity of a differentiable function. Rules for calculation of derivatives. Vanishing of the derivative at a local extremum of a differentiable function, Rolle's theorem and Mean Value Theorem. Values of the derivative and behavior of the function.

Integrals The Darboux integral of a bounded function. The test for integrability. Integrability of monotonic and continuous functions. Properties of Darboux integrals, Intermediate Value theorem for integrals. Fundamental theorem of Calculus. Integration by parts and theorem on change of variable. Term-wise differentiation and integration of power series. Taylor series for a function.

Disabilities

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the Evacuation Guide for People with Physical Disabilities.