# Introduction to Real Analysis (MATH351) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Introduction to Real Analysis MATH351 5. Semester 4 0 0 4 7
Pre-requisite Course(s)
Math 136
Course Language English Compulsory Departmental Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Discussion, Question and Answer, Problem Solving. Assoc. Prof. Dr. Erdal KARAPINAR The aim of the course is providing a familiarity to concepts of the real analysis, such as, limit, continuity, differentiation, connectedness, compactness, convergence etc. The students who succeeded in this course; At the end of the course the students are expected to: 1) know real numbers and real number system, 2) know countable, uncountable, finite sets, 3) know sequences of real numbers (Cauchy Sequences) 4) know uniform convergence of sequences of functions 5) know metric spaces 6) know compactness and connectedness, Contraction Mapping Theorem, 7) know Arzela-Ascoli Theorem, Extension Theorem of Tietze, Baire’s Theorem. A review of sets and functions, real numbers (or system), countable and uncountable sets, sequences of real Numbers (Cauchy sequences), Uniform Convergence of Sequences of functions, Metric Spaces, Compactness and Connectedness, Contraction Mapping Theorem, Arzela-Ascoli Theorem, Extension Theorem fo Tietze, Baire?s Theorem.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Real Number System: Axioms, Some Consequences of the Least Upper Bound Property Read the related pages in the text book
2 Absolute value and intervals, Sequences of Real Numbers Read the related pages in the text book
3 Accumulation Points: Theorems of Bolzano and Weierstras Read the related pages in the text book
4 Limit Superior and Inferior Read the related pages in the text book
5 Metric Spaces: Examples, Open and Closed Subsets Read the related pages in the text book
6 Sequences in a Metric Space Read the related pages in the text book
7 Midterm Exam
8 Contiunity of Functions, Cartesian Product of Metric Spaces Read the related pages in the text book
9 Completion of a Metric Space Read the related pages in the text book
10 Compactness and Connectedness: Compact Sets, Compactness and Convergence of Sequences Read the related pages in the text book
11 Continuity and Compactness, Connectedness Read the related pages in the text book
12 Connected Components Read the related pages in the text book
13 Applications: Contraction Mapping Theorem Read the related pages in the text book
14 The Arzela-Ascoli Theorem, Extension Theorem of Tietze Read the related pages in the text book
15 Baire’s Theorem Read the related pages in the text book
16 Final

### Sources

Other Sources 1. An introduction to Real Analysis, T. Terzioğlu, Matematik Vakfı. 2. Real Analysis, H. L. Royden, Prentice-Hall 3. Principles of Mathematical Analysis, W. Rudin, 3rd Edition 1976, McGraw-Hill Inter. Edit.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments - -
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 60
Final Exam/Final Jury 1 40
Toplam 3 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class
Presentation/Seminar Prepration
Project
Report
Homework Assignments
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury
Prepration of Final Exams/Final Jury