Real Analysis (AN1)

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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.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Individual study 110 Exercises solving 38.5 Course total 187.5
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.