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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 3053 REAL ANALYSIS I ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

PROFESSOR DOCTOR SELÇUK DEMIR

Offered to

Mathematics (Evening) Mathematics

Course Objective

The aim of the course is to prepare a backround for modern analysis and for the other branches which use these theories : Metric Spaces, Completion of a Metric Space, Continuity, Compactness and Connectedness on Metric Space, Contraction Mapping Theorem and its Applications, The Arzela-Ascoli Theorem, Peona Theorem, The Tietze Extension Theorem. Baire's Theorem

Learning Outcomes of the Course Unit

1   will be able to define metric spaces 2   will be able to relate Continuity , Compactness and Connectedness 3   will be able to use the Banach Fixed Point Theorem 4   will be able to use the Arzela-Ascoli Theorem 5   will be able to understand the Tietze Extension Theorem 6   will be able to understand the Baire's Theorem 7   will be able to write complete and formal proofs for the problems related with the above topics.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Limit Superior and Limit Inferior. 2 Metric Spaces. The l_p sequence spaces 3 Continuity on Metric Spaces. Completion of a Metric Space. 4 Compactness, Connectedness and Continuity. 5 Cartesian product of Metric Spaces. 6 B(X); the space of bounded real valued functions on X and C(X); the space of continuous real valued functions on X. 7 Banach Fixed Point Theorem. 8 Midterm 9 Application of Banach Fixed Point Theorem to Linear Equations, Differential Equations and Integral Equations. 10 Schauder Fixed Point Theorem. 11 Arzela-Ascoli Theorem. 12 Peano Theorem. 13 Extension Theorem. 14 Baire's Theorem.

Recomended or Required Reading

Textbook(s): An Introduction to Real Analysis; T.Terzioglu, 1994, METU. Supplementary Book(s): Real Analysis, 3rd Edition; H.L. Royden, Macmillan Publishing Company.

Planned Learning Activities and Teaching Methods

Lecture Notes Text Book(s) Solving Problems

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.40 + ASG * 0.20 + FIN * 0.40 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: sedef.erim@deu.edu.tr Office: (232) 3018591

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 4 52 Preparations before/after weekly lectures 12 4 48 Preparation for midterm exam 1 20 20 Preparation for final exam 1 30 30 Preparing assignments 2 7 14 Final 1 2 2 Midterm 1 2 2 TOTAL WORKLOAD (hours) 168

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 PO.12 PO.13 LO.1 5 5 4 4 4 3 4 LO.2 5 5 4 4 4 3 4 LO.3 5 5 4 4 4 3 4 3 LO.4 5 5 4 4 4 3 4 3 LO.5 5 5 4 4 4 3 4 LO.6 5 5 4 4 4 3 4 LO.7 5 5 4 4 4 3 4 3