Real Analysis (MAE617)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE511
Semester	5
Course Title	Real Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	http://www.math.uoi.gr/GR/studies/undergraduate/courses/perigr/MAE_511.pdf

Learning Outcomes

Learning outcomes	The course aims in presenting topics concerning real valued functions defined on a metric space. Pointwise and uniform convergence of a sequence of functions are discussed as so as topics like Ascoli-Arzela theorem and Stone-Weirstrass theorem. Applications of the above are also given.
General Competences	Working independently Team work Working in an international environment Working in an interdisciplinary environment Production of new research ideas.

Syllabus

Function spaces on a metric space (X,d), pointwise and uniform convergence of sequence of functions, the space B(X) of real bounded functions on X-, the space C(X) of continuous functions on X – equicontinuous subsets of C(X), Ascoli-Arzela theorem and applications, Dini's theorem, Stone-Weierstrass theorem and applications, separable metric spaces, Lindelof's theorem on Euclidean spaces, the Cantor set, the Cantor function-applications.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT for the presentation and communication for submission of the exercises
Teaching Methods	Activity Semester Workload Lectures 39 Home exercises 30 Independent study 81 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See Eudoxus. Additionally:

• Charalambos D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, Academic Press.
• Michael O Searcoid, Metric Spaces, Springer Undergraduate Mathematics Series.