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Course Description Course Name : Analysis Course Code : MT-572 Course Type : Compulsory Level of Course : Second Cycle Year of Study : 1 Course Semester : Spring (16 Weeks) ECTS : 6 Name of Lecturer(s) : Assoc.Prof.Dr. ALİ ARSLAN ÖZKURT Learning Outcomes of the Course : understands the measure theory knows lebesgue integration of real and complex functions knows Riezs representation theorem which is one of the important theory of functional analysis and some results of it learns Lebesgue measure in Euclidean spaces learns Banach and L^p spaces knows Hahn-Banach theorem which is one of the important theory of Real analysis Mode of Delivery : Face-to-Face Prerequisites and Co-Prerequisites : None Recommended Optional Programme Components : None Aim(s) of Course : to give basic knowledges of measure theory, to give basic knowledges of Lebesque measure and Lebesgue integration of real and complex functions, to give the basic theorems of real and functional analysis like Riezs representation theorem and Hahn-Banach theorem Course Contents : Measure ,Step functions and simple functions , integral of positive and complex valued functions,Topologicl preliminaries , Riezs representation theorem, Borel measures, Lebesque measure, continuity properties of measurable functions, convex fonctions and some inequalities, L^p spaces, Banach spaces, Baire´s theorem and consequences, Hahn-Banach theorem. Language of Instruction : Turkish Work Place : Classroom   Course Outline /Schedule (Weekly) Planned Learning Activities Week Subject Student's Preliminary Work Learning Activities and Teaching Methods 1 definition of measures and elementary properties of measures Study the relevant sections in the textbook and solve problems Lecture and discussion 2 Step and simple functions Study the relevant sections in the textbook and solve problems Lecture and discussion 3 Integration of positive functions and integration and complex functions Study the relevant sections in the textbook and solve problems Lecture and discussion 4 Topological preliminaries (Urysohn lemma and partition of unity) Study the relevant sections in the textbook and solve problems Lecture and discussion 5 Riezs representation theorem Study the relevant sections in the textbook and solve problems Lecture and discussion 6 Regularity properties of Borel measures Study the relevant sections in the textbook and solve problems Lecture and discussion 7 Lebesgue measures in Euclidean spaces Study the relevant sections in the textbook and solve problems Lecture and discussion 8 midterm-exam Reviewed of the topics discussed in the lecture notes and source again written examination 9 Continuity properties of measurable functions Study the relevant sections in the textbook and solve problems Lecture and discussion 10 convex functions and some inequalities Study the relevant sections in the textbook and solve problems Lecture and discussion 11 L^p spaces Study the relevant sections in the textbook and solve problems Lecture and discussion 12 Banach spaces Study the relevant sections in the textbook and solve problems Lecture and discussion 13 Baire´s theorem and consequences of Baire´s theorem Study the relevant sections in the textbook and solve problems Lecture and discussion 14 Hahn-Banach theorem Study the relevant sections in the textbook and solve problems Lecture and discussion 15 Radon-Nikodym theorem Study the relevant sections in the textbook and solve problems Lecture and discussion 16/17 final-exam Revision of the topics discussed in the lecture notes and sources written examination   Required Course Resources Resource Type Resource Name Recommended Course Material(s)  Real and Complex Analysis, W Rudin  Principles of Real Analysis, D. Aliprantis, O. Burkinshaw Required Course Material(s)   Assessment Methods and Assessment Criteria Semester/Year Assessments Number Contribution Percentage     Mid-term Exams (Written, Oral, etc.) 1 60     Homeworks/Projects/Others 1 40 Total 100 Rate of Semester/Year Assessments to Success 40   Final Assessments 100 Rate of Final Assessments to Success 60 Total 100   Contribution of the Course to Key Learning Outcomes # Key Learning Outcome Contribution* 1 Aquires sufficient knowledge to enable one to do research over and above the undergraduate level 4 2 Learns theoretical foundations of his/her field thoroughly 5 3 Uses the knowledge in his/her field to solve mathematical problems 4 4 Proves basic theorems in different areas of Mathematics 3 5 Creates a model for the problems in his/her field and expresses the relationship between objects in a simple and comprehensive way. 3 6 Uses technical tools in his/her field 4 7 Works independently in his/her field requiring expertise 4 8 Works with other colleagues collaboratelly, takes responsibilities if necessary during the research process 4 9 Argues and analyzes knowledge in his/her field and applies them in other fields if necessary 4 10 Has sufficient knowledge to follow literature in his/her field and communicate with stakeholders 4 11 Uses the language and technology to develop knowledge in his/her field and shares these with stakeholders systematically when necessary 1 12 Knows and abides by the ethical rules in analyzing, solving problems and publishing results 5 * Contribution levels are between 0 (not) and 5 (maximum).   Student Workload - ECTS Works Number Time (Hour) Total Workload (Hour) Course Related Works     Class Time (Exam weeks are excluded) 14 4 56     Out of Class Study (Preliminary Work, Practice) 14 4 56 Assesment Related Works     Homeworks, Projects, Others 1 3 3     Mid-term Exams (Written, Oral, etc.) 1 15 15     Final Exam 1 20 20 Total Workload: 150 Total Workload / 25 (h): 6 ECTS Credit: 6