Introduction to Real Analysis

Course Code:

MATH 256

Course Period:

Spring

Course Type:

Core

P:

3

Lab:

2

Credits:

4

ECTS:

8

Course Language:

İngilizce

Course Objectives:

Learning the basic concepts and results of real analysis in Rp. Introducing the elementary topological concepts in Rp

Course Content:

Elements of point set theory.Functions and the Real Number System. Open and closed sets in Rp. Accumulation points. Bolzano-Weierstrass theorem in Rp. Compactness and connectedness.Heine Borel Theorem.Convergence of sequences in Rp. Sequences of Functions, Uniform Convergence Continuity and Uniform continuity.Sequences of Continuous Functions,Limits of Functions.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Teaching Methods	Assessment Methods
1) Knows the topology of Euclidean spaces.	1-2	A-B
2) Knows the concept of convergence of sequences and related important teorems.	1-2	A-B
3) Knows local and global properties of continuous functions.	1-2	A-B

Course Flow

Week	Topics	Study Materials
1	Introduction, Functions, the Real Number System
2	Suprema, Infima and Supremum Principle
3	Vector Spaces, Inner Product Spaces, Schwarz Inequality, The Space Rp
4	Open Sets, Nested Intervals Theorem,
5	Accumulation Points, Bolzano- Weierstrass Theorem, Connected Sets
6	Compactness, The Heine-Borel Theorem
7	Sequences in Rp, Convergence
8	Monotone Convergence Theorem, Bolzano-Weierstrass Theorem
9	Cauchy Sequences, Cauchy Convergence Criterion
10	Sequences of Functions, Convergence, Uniform Convergence, Uniform Norm, Cauchy Criterion for Uniform Convergence
11	Continuous Functions, Local Properties of Continuous Functions
12	Global Properties of Continuous Functions
13	Sequences of Continuous Functions
14	Limits of Functions

Recommended Sources

Textbook	The Elements of Real Analysis, Robert G. Bartle
Additional Resources

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	80
Quizzes	5	10
Assignments	5	10
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		50
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		50
Total		100

COURSE CATEGORY

Expertise/Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)	x
2	Acquiring fundamental knowledge on fundamental research fields in mathematics				x
3	Ability form and interpret the relations between research topics in mathematics			x
4	Ability to define, formulate and solve mathematical problems		x
5	Consciousness of professional ethics and responsibilty				x
6	Ability to communicate actively			x
7	Ability of self-development in fields of interest			x
8	Ability to learn, choose and use necessary information technologies	1
9	Lifelong education			x

ECTS

Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	5	70
Hours for off-the-classroom study (Pre-study, practice)	14	5	70
Mid-terms (Including self study)	2	10	20
Quizzes	5	2	10
Assignments	5	2	10
Final examination (Including self study)	1	15	15
Total Work Load			195
Total Work Load / 25 (h)			7.8
ECTS Credit of the Course			8

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