Course Information
SemesterCourse Unit CodeCourse Unit TitleL+PCreditNumber of ECTS Credits
4MATH252ANALYSIS4+046

Course Details
Language of Instruction English
Level of Course Unit First Cycle
Department / Program MATHEMATICS
Mode of Delivery Face to Face
Type of Course Unit Compulsory
Objectives of the Course This course is designed to develop the intuitive understanding and theory skills necessary for the concepts of real analysis. Upon completion of the course, students will have a working knowledge of the fundamental definitions and theorems of real analysis .
Course Content Real numbers, completeness of R, convergence of sequences, Cauchy sequences, open and closed sets, compact sets, Heine-Borel and Bolzano-Weierstrass Theorems, connected and path-connected sets, continuous mappings, function sequences, pointwise and uniform convergence, function series, the inverse function theorem, implicit functions and the implicit function theorem.
Course Methods and Techniques
Prerequisites and co-requisities None
Course Coordinator None
Name of Lecturers Prof.Dr. Ba┼čak AY SAYLAM
Assistants None
Work Placement(s) No

Recommended or Required Reading
Resources Elementary Classical Analysis, J. E. Marsden.
Introduction to Real Analysis, R. G. Bartle.

Course Category

Planned Learning Activities and Teaching Methods
Activities are given in detail in the section of "Assessment Methods and Criteria" and "Workload Calculation"

Assessment Methods and Criteria
In-Term Studies Quantity Percentage
Midterm exams 2 % 40
Quizzes 2 % 10
Homeworks 0 % 0
Other activities 0 % 0
Laboratory works 0 % 0
Projects 0 % 0
Final examination 1 % 50
Total
5
% 100

ECTS Allocated Based on Student Workload
Activities Quantity Duration Total Work Load
Weekly Course Time 1 42 42
Outside Activities About Course (Attendance, Presentation, Midterm exam,Final exam, Quiz etc.) 1 6 6
Application (Homework, Reading, Self Study etc.) 1 24 24
Exams and Exam Preparations 1 67 67
Total Work Load   Number of ECTS Credits 5 139

Course Learning Outcomes: Upon the successful completion of this course, students will be able to:
NoLearning Outcomes
1 The ability to determine whether a set is open or closed.
2 To be able to determine the closure and the boundary of a set.
3 To understand the definitions of a compact set and a connected set.
4 To understand continuous functions and that they are bounded on compact sets.
5 To understand the definitions of pointwise and uniform convergence.
6 To be able to apply the Weierstrass M-test, Abel test and Drichlet test to determine uniform convergence of series..


Weekly Detailed Course Contents
WeekTopicsStudy MaterialsMaterials
1 Ordered fields and the number system, completeness and the real number system. Elementary Classical Analysis, J. E. Marsden.
2 Least upper bounds, Cauchy sequences. Elementary Classical Analysis, J. E. Marsden.
3 Cluster points; lim inf and lim sup, Euclidean space. Elementary Classical Analysis, J. E. Marsden.
4 Open sets, interior of a set, closed sets, accumulation points. Elementary Classical Analysis, J. E. Marsden.
5 Closure of a set, boundary of a set, sequences, series in R and R^n. Elementary Classical Analysis, J. E. Marsden.
6 Compact sets; the Heine-Borel and the Bolzano-Weierstrass Theorems, nested set theory. Elementary Classical Analysis, J. E. Marsden.
7 Path-connected sets, connected sets. Elementary Classical Analysis, J. E. Marsden.
8 Continuity, images of compact and connected sets. Elementary Classical Analysis, J. E. Marsden.
9 Operations on contionuous mappings, the boundedness of continuous functions on compact sets. Elementary Classical Analysis, J. E. Marsden.
10 The Intermediate-Value theorem, uniform continuity. Elementary Classical Analysis, J. E. Marsden.
11 Pointwise and uniform convergence, the Weierstrass M-test, integration and differentiation of series. Elementary Classical Analysis, J. E. Marsden.
12 The space of continuous functions,The Arzoli-Ascoli Theorem, fixed points and integral equations. Elementary Classical Analysis, J. E. Marsden.
13 Stone-Weierstrass Theorem, the Drichlet and Abel Tests, Mean-Value and Taylor Theorems. Elementary Classical Analysis, J. E. Marsden.
14 Inverse function theorem, Implicit function theorem. Elementary Classical Analysis, J. E. Marsden.


Contribution of Learning Outcomes to Programme Outcomes
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14
C1 3 3 3 3 3
C2 3 3 3 3 3
C3 3 3 3 3 3
C4 3 3 3 3 3
C5 3 3 3 3 3
C6 3 3 3 3 3

Contribution: 0: Null 1:Slight 2:Moderate 3:Significant 4:Very Significant


https://obs.iyte.edu.tr/oibs/bologna/progCourseDetails.aspx?curCourse=163239&lang=en