Real Analysis (AN1)
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- Graduate Courses Outlines
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General
| School | School of Science |
|---|---|
| Academic Unit | Department of Mathematics |
| Level of Studies | Graduate |
| Course Code | AN1 |
| Semester | 1 |
| Course Title | Real Analysis |
| Independent Teaching Activities | Lectures (Weekly Teaching Hours: 3, Credits: 7.5) |
| Course Type | General Background |
| Prerequisite Courses |
Introduction to Topology |
| Language of Instruction and Examinations |
Greek |
| Is the Course Offered to Erasmus Students | Yes (in English) |
| Course Website (URL) | See eCourse, the Learning Management System maintained by the University of Ioannina. |
Learning Outcomes
| Learning outcomes |
The plan of the course is the deeper study of the theory of metric spaces. The Stone - Weirstrass theorem is presented and also there are studied theorems that involve families of equicontinuous functions. Among others there are studied the following topics: the Cantor set, totally bounded and compact metric spaces, the Hausdorff metric and the Tietze theorem. Moreover, applications of the above theorems are given. |
|---|---|
| General Competences |
The objective of the course is the graduate student’s ability achievement in analysis and synthesis of deeper knowledge of Real Analysis. |
Syllabus
The Ascoli - Arzela and Stone - Weirstrass theorems and applications, the Cantor set, characterization of totally bounded metric spaces via subsets of Cantor set, extensions of continuous functions and the Tietze theorem, the space S(X) of closed and bounded subsets of a metric space and the metric Hausdorff on S(X), characterization of completeness of the metric space S(X) equipped with the metric Hausdorff and applications, the selection Blashke theorem, applications of the fixed point theorem of Banach, partitions of unity.
Teaching and Learning Methods - Evaluation
| Delivery |
Teaching with talks on the blackboard. | ||||||||||
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| Use of Information and Communications Technology | - | ||||||||||
| Teaching Methods |
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| Student Performance Evaluation |
Final written exam or student's presentations on the blackboard. The student can choose either of the above ways of examination or both with final grade the higher one. |
Attached Bibliography
- Charalambos D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, Academic Press.
- Michael O Searcoid, Metric Spaces, Springer Undergraduate Mathematics Series.