MATHEMATICS
Bachelor TR-NQF-HE: Level 6 QF-EHEA: First Cycle EQF-LLL: Level 6

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT3001 Real Analysis I Fall 3 0 3 5

Basic information

Language of instruction: English
Type of course: Must Course
Course Level: Bachelor’s Degree (First Cycle)
Mode of Delivery: Face to face
Course Coordinator : Instructor MOHAMED KHALIFA
Recommended Optional Program Components: There is none.
Course Objectives: The aim of the course is to give to the student to learn enough examples, theorems and techniques in analysis to be well prepared for the standart graduate courses in topology, measure theory and functional analysis.

Learning Outcomes

The students who have succeeded in this course;
Able to use the basic examples, theorems and techniques for the standart graduate courses in topology, measure theory and functional analysis

Course Content

The topology of IR, Cauchy sequences, The concepts of lower and upper limit, General topics on functions, Concepts of equivalent metrics, Complete metric spaces, Contractions, Normed vector spaces, Banach spaces, Fixed point theorem and its applications, Orthonormal spaces and introduction to Hilbert spaces.

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Topology of real numbers
2) Cauchy sequences
3) Concepts of upper and lower limits
4) General topics on functions
5) Equivalent metrics
6) Complete metric spaces
7) Contraction mappings
8) Normed vector spaces
9) Normed vector spaces
10) Banach spaces
11) Banach spaces
12) Fixed point theory and applications
13) Orthonormal spaces and introduction to the Hilbert spaces
14) Orthonormal spaces and introduction to the Hilbert spaces

Sources

Course Notes / Textbooks: Royden, H. L. “Real Analysis”, 3rd ed. MacMillan Publishing Company (1988)
References: .

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Quizzes 5 % 15
Midterms 2 % 45
Final 1 % 40
Total % 100
PERCENTAGE OF SEMESTER WORK % 60
PERCENTAGE OF FINAL WORK % 40
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 3 42
Study Hours Out of Class 14 1 14
Quizzes 5 1 5
Midterms 2 10 20
Final 1 19 19
Total Workload 100

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
           
Program Outcomes Level of Contribution
1) To have a grasp of basic mathematics, applied mathematics and theories and applications in Mathematics 5
2) To be able to understand and assess mathematical proofs and construct appropriate proofs of their own and also define and analyze problems and to find solutions based on scientific methods, 5
3) To be able to apply mathematics in real life with interdisciplinary approach and to discover their potentials, 5
4) To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself, 4
5) To be able to tell theoretical and technical information easily to both experts in detail and non-experts in basic and comprehensible way, 4
6) To be familiar with computer programs used in the fields of mathematics and to be able to use at least one of them effectively at the European Computer Driving Licence Advanced Level,
7) To be able to behave in accordance with social, scientific and ethical values in each step of the projects involved and to be able to introduce and apply projects in terms of civic engagement,
8) To be able to evaluate all processes effectively and to have enough awareness about quality management by being conscious and having intellectual background in the universal sense,
9) By having a way of abstract thinking, to be able to connect concrete events and to transfer solutions, to be able to design experiments, collect data, and analyze results by scientific methods and to interfere, 5
10) To be able to continue lifelong learning by renewing the knowledge, the abilities and the competencies which have been developed during the program, and being conscious about lifelong learning,
11) To be able to adapt and transfer the knowledge gained in the areas of mathematics ; such as algebra, analysis, number theory, mathematical logic, geometry and topology to the level of secondary school, 3
12) To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively. 4