Course Information

Course Name	Turkish	: İleri Matematik
	English	Advanced Mathematics
Course Code	MAT 272E	Credit	Lecture (hour/week)	Recitation (hour/week)	Laboratory (hour/week)
Semester	-	3	3	-	-
Course Language		English
Course Coordinator		İbrahim Kırat
Course Objectives		1. To teach the student the techniques and methods of Mathematical Analysis and to allow the student to develop a certain level of proficiency in these methods. 2. To teach students to use the basic concepts they learned in Calculus classes in a mathematically rigourous way.
Course Description		Sequences and series of real numbers and convergence. Finite dimensional real vector spaces. Young’s, Hölder’s and Minkowski’s inequalities. Metric spaces. Sequences in metric spaces. Convergence and boundedness. Cauchy sequences and completeness. Topology of Metric spaces: open and closed sets. Compactness. Heine-Borel Theorem. Real valued continuous functions on metric spaces and their metric structure. Continuity and uniform continuity. Lipschitz continuity. Total Derivative. C^k[a,b] and \ell^p spaces. Sequences and series of real valued functions on metric spaces. Pointwise and uniform convergence. Cauchy criterion for uniform convergence. Weierstrass M-test. The Stone-Weierstrass Theorem. Hilbert spaces.
Course Outcomes		Students completing this course will be able to: I. Know the proofs of limit theorems on metric spaces and use these theorems in calculations correctly, II. Know the proofs of theorems on bounded, monotone, continuous and uniformly continuous functions and use these theorems, III. Determine pointwise convergence or uniform convergence in sequences and series of functions IV. Know the basic concepts of the Lebegue integration theory for the functions defined on real numbers.
Pre-requisite(s)
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Textbook		1. W. R. Parzynski and P. W. Zipse, Introduction to Mathematical Analysis, McGraw-Hill Book Company, 1987. 2. Erwin O. Kreyszig, Introductory Functional Analysis with Applications, Wiley; 1-st edition, 1989. 3. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976 4. Micheal O. Searcoid, Metric Spaces, Springer, 2007. 5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons; 3-rd edition, 2000.
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