Course Title:                       Introduction to Real Analysis II

Course Code:                      MATH 3277

Level:                                  3

Number of Credits:              3

Semester:                            2

Prerequisite(s):                    MATH 2277

Course Rationale

This course serves to complete the foundation in real analysis required for a major in mathematics, and is designed to prepare students for advanced courses in analysis and functional analysis. It will also serve as a useful platform for further study in complex analysis, topology, dynamical systems, quantum mechanics, and mathematical statistics. The rigorous treatment of the subject (in terms of theory and examples) will provide students with an opportunity to enhance their capacity for mathematical reasoning and intuition.

Course Description

This is the follow-up course for MATH 2XXX Introduction to Real Analysis I. The course exposes students to rigorous mathematical definitions, proofs and classical results on differentiation, Riemann integration, sequences and series of functions. Major emphasis is placed on the proper use of definitions for the rigorous proof of theorems.  The following topics will be covered: Differentiation, Riemann integration, sequences and series of functions and metric spaces.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly problem sheets, allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures, and periodic marked assignments. Assessment will be based on the marked assignments and in-course tests followed by a final examination based on the whole course

Learning Outcomes

By the end of the course, students will be able to:

• Provide and utilize the concept of the derivative, L'Hôpital's rule, Taylor’s formula.
• Define and expound upon the following concepts: Upper sum, lower sum, Riemann integrability.
• Prove the major theorems of analysis of the real line
• Prove that certain sequences of functions converge uniformly.
• Prove that various sequences are convergent and/or Cauchy in a metric space.
• Prove theorems regarding compactness and connectedness in a metric space.
• Construct counter-examples to show to show that certain hypotheses for classical theorems of real analysis are necessary.
• Prove that certain functions are continuous in a metric space.

Course Contents

1. Differentiation: Derivatives, Rolle's theorem. Mean value theorem, Darboux's theorem on intermediate value property of derivatives. Taylor's theorem.
2. Integration: The Riemann Integral and its properties. Integrability of continuous and monotonic functions. The fundamental theorem of Calculus. Mean value theorems of integral Calculus. Improper Integrals.
3. Sequence and series of functions: Point wise convergence. Uniform convergence and its relation to continuity, integration, and differentiation. Weierstrass M-test.
4. Metric spaces: Definition and examples, neighbourhoods, limit points, interior, and boundary points. Open and closed sets. Closure, interior, and boundary of a set. Subspaces.
5. Sequences in Metric Space: Cauchy sequences and complete spaces. Cantor's intersection theorem and the contraction mapping principle. Dense and nowhere dense subsets.
6. Compactness in Metric Space: Sequential compactness and Heine-Borel property, finite intersection property, continuous functions on compact sets.

Teaching Methodology

Students will be exposed to the theoretical aspects of the mathematical analysis through informative lectures. Tutorials provided as needed during scheduled lecture hours will reinforce their learning and provide experience with the practical application of different type of problems.

The total estimated 39 contact hours may be accounted for as follows: 26 hours of lectures and 13 hours of tutorials. Course material, including practice problems, will be posted on myelearning.

Assessment

The course assessment has three components:

Final exam:  2-hour written paper                       50%

Two Midterm Exams     (15% each)                      30%

Five Written Assignments (4% each)                   20%

Course Calendar

Week

Lecture Topics

Assignments

Tutorial

1

Course Overview/Introduction

Introduction: Derivability of a function at a point, Derivability on an interval, Derivability and continuity of a function, Caratheodory’s theorem, Algebra of derivatives,

None

Tutorial #1

2

Differentiation: Derivatives, Rolle’s theorem. Mean Value Theorem, Darboux's theorem on intermediate value property of derivatives. Taylor's theorem

Assignment 1 given

Tutorial #2

3

Riemann Integration: The Riemann Integral and its properties. Integrability of continuous and monotonic functions.

None

Tutorial #3

4

Riemann Integration (continued) The fundamental theorem of calculus. Mean value theorems of integral calculus. Improper Integrals.

Assignment 1 due

Assignment 2 given

Tutorial #4

5

Sequence and series of functions: Point wise convergence. Uniform convergence, uniform convergence and continuity.

Coursework Exam #1

(15%)

No tutorial this week

6

Sequence and series of functions (continued): Uniform convergence & integration, and Uniform convergence & differentiation. Weierstrass M-test.

Assignment 2 due

Assignment 3 given

Tutorial #5

7

Metric spaces: Definition and examples, Open balls / neighbourhoods, limit points, interior, and boundary points. Examples in R, Rn, and arbitrary spaces

None

Tutorial #6

8

Metric spaces (Continued) : Open and closed sets. Closure, interior, and boundary of a set. Subspaces, Examples in R, Rn, and arbitrary spaces.

Assignment 3 due

Assignment 4 given

Tutorial #7

9

Sequences in Metric Space: Cauchy sequences and complete spaces. Cantor's intersection theorem. Examples in R, Rn, and arbitrary spaces.

None

Tutorial #8

10

Sequences in Metric Space (Continued) Banach fixed point theorem /the contraction mapping principle. Dense and nowhere dense subsets. Baire Category Theorem.

Assignment

4 due

Tutorial #9

11

Compactness in Metric Space: Sequential compactness and Heine-Borel property. Examples in R, Rn, and arbitrary spaces.

Coursework Exam #2

(15%)

No Tutorial this week

12

Compactness in Metric Space (Continued): Totally bounded spaces, finite intersection property, and continuous functions on compact sets. Connectedness in metrics space. Examples in R, Rn, and arbitrary spaces.

None

Tutorial #10

13

Revision

None

None

Reference Material

Books:

Prescribed

The required text is Analysis, with an Introduction to Proof, by Steven R. Lay, Fourth          Edition, Pearson (Prentice Hall) 2005.

Recommended

Bartle, Robert G. &Sherbert, Donald R., INTRODUCTION TO REAL ANALYSIS, third            ed., John Wiley & Sons, Inc., 2000.

‘Methods of Real Analysis’ by Richard R. Goldberg.-Oxford and IBH Publishing co.

LANG, Serge. Undergraduate Analysis (Undergraduate Texts in Mathematics), second edition, Springer, 2010.ISBN: 0387948414.

These books are pedagogically sound, comprehensively address all element of the syllabus, and provide useful case studies and examples.

ONLINE RESOURCES: