CANKIRI KARATEKIN UNIVERSITY

Bologna Information System

Course Title | Code | Semester | Laboratory+Practice (Hour) | Pool | Type | ECTS |

REAL ANALYSIS | MAT404 | SPRING | 4+0 | C | 8 |

Learning Outcomes | 1-To determine the measurability of a set. 2-To express the properties of measure function. 3-To compute the integral of a measurable simple function. 4-To explain the cases for which riemann integral is insufficient and the usefulness of lebesgue integral. 5- to explain convergence in l_p and in measure, the relation between these convergences. |

Activity | Percentage (100) | Number | Time (Hours) | Total Workload (hours) |

Course Duration (Weeks x Course Hours) | 14 | 4 | 56 | |

Classroom study (Pre-study, practice) | 14 | 6 | 84 | |

Assignments | 10 | 2 | 14 | 28 |

Short-Term Exams (exam + preparation) | 10 | 2 | 14 | 28 |

Midterm exams (exam + preparation) | 30 | 1 | 16 | 16 |

Project | 0 | 0 | 0 | 0 |

Laboratory | 0 | 0 | 0 | 0 |

Final exam (exam + preparation) | 50 | 1 | 18 | 18 |

Other | 0 | 0 | 0 | 0 |

Total Workload (hours) | 230 | |||

Total Workload (hours) / 30 (s) | 7,67 ---- (8) | |||

ECTS Credit | 8 |

Week | Topics | Study Metarials |

1 | Preliminaries, countable and uncountable sets, functions, sequences, convergence of sequences | |

2 | bounded sequences, sequence of sets and their limits, some set classes, sigma Ring and sigma algebras | |

3 | Borel algebras, measurable set, measure function and its properties | |

4 | outer measure and Lebesgue outer measure, Lebesgue measure | |

5 | measurable functions, generating measurable functions from measurable functions | |

6 | integral of simple functions, integral of positive functions | |

7 | Monotone Convergence Theorem, Fatou lemma, Beppo-Levi Theorem | |

8 | Lebesgue integral, Absolute integrability property of Lebesgue integral | |

9 | Tchebichev inequality, Charge function and its properties Lebesgue co | |

10 | nvergence theorem and its applications, Bounded Convergence Theorem and its applications | |

11 | The relation between Riemannian integral and Lebesgue integral | |

12 | L_p spaces and its properties | |

13 | Hölder and Minkowski inequalities, Riesz-Fischer Theorem | |

14 | L infinity spaces and its properties, uniform convergence, convergence in measure and Lp- convergence. |

Prerequisites | - |

Language of Instruction | Turkish |

Course Coordinator | Assist. Prof. Dr. Gülsüm ULUSOY |

Instructors | - |

Assistants | - |

Resources | Introduction to real analysis, M. Stoll, Addison Wesley, 2000. |

Supplementary Book | [1] Reel Analiz, Mustafa Balcı, Ertem Matbaası, 2000. [2] Measure, Integral and Probability, Capinski, M . ve Kopp, E. Springer,1999. [3] Lebesgue Measure and Integration, P. K. Jain and V. P. Gupta, John Willey and Sons Inc., 1996. [4] Real analysis with Real Applications, K. R. Davidson, A. P. Donsig, Prentice Hall, 2002. |

Document | Lecture notes. |

Goals | To analyze properties of measure theory, Lebesgue integral and L_p spaces in real numbers set. |

Content | - |

Program Learning Outcomes | Level of Contribution | |

1 | To have a grasp of theoretical and applied knowledge in main fields of mathematics | 5 |

2 | To have the ability of abstract thinking | 4 |

3 | To be able to use the gained mathematical knowledge in the process of identifying the problem, analyzing and determining the solution steps | 5 |

4 | To be able to relate the gained mathematical acquisitions with different disciplines and apply in real life | 2 |

5 | To have the qualification of studying independently in a problem or a project requiring mathematical knowledge | 3 |

6 | To be able to work compatibly and effectively in national and international groups and take responsibility | - |

7 | To be able to consider the knowledge gained from different fields of mathematics with a critical approach and have the ability to improve the knowledge | 5 |

8 | To be able to determine what sort of knowledge the problem met requires and guide the process of learning this knowledge | 4 |

9 | To adopt the necessity of learning constantly by observing the improvement of scientific accumulation over time | 3 |

10 | To be able to transfer thoughts on issues related to mathematics, proposals for solutions to the problems to the expert and non-expert shareholders written and verbally | 4 |

11 | To be able to produce projects and arrange activities with awareness of social responsibility | - |

12 | To be able to follow publications in mathematics and exchange information with colleagues by mastering a foreign language at least European Language Portfolio B1 General Level | - |

13 | To be able to make use of the necessary computer softwares (at least European Computer Driving Licence Advanced Level), information and communication technologies for mathematical problem solving, transfer of thoughts and results | - |

14 | To have the awareness of acting compatible with social, scientific, cultural and ethical values | - |

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