Math 349 INTRODUCTION TO MATHEMATICAL ANALYSIS

Course Number and Title: 2360349, Introduction to Mathematical Analysis

METU Credit & ECTS Credit: (4-0)4 and 9.0
Prerequisite Courses: Math 252

Catalogue Description: LUB Property of real numbers. Compactness, connectedness, limits and continuity in metric spaces. Sequences and series of scalars, complete metric spaces, limsup. Sequences and series of functions, uniform convergence, applications.


Course Objectives: The students will learn the basic results in the field and get expirience in standard methods of real analysis.
The students will get acquainted with the basic notions of the real number system (axioms, convergence, limit superior/inferior of real sequences, etc.).
The students will get acquainted with the basic notions of metric spaces (topological properties, continuity/equicontinuity of functions, products, completion, compactness, connectedness, contraction mappings principles and their applications, Tietze extension theorem, Baire's theorem, etc.)

Textbooks: T. Terzioğlu: An Introduction to Real Analysis. Matematik Vakfı, 1999.
               Walter Rudin - Principles of Mathematical Analysis, Third Edition-McGraw-Hill Science Engineering Math (1976)


Course Outline (14 weeks):
Week                 Topics   
  1           Real Number System: axioms; LUB property; real sequences.
  2           Real Number System: Bolzano - Weierstrass theorem; limit superior/inferior.
  3           Metric Spaces: subspaces, open and closed subsets of a metric space, closure of a subset, examples of metric spaces.
  4           Metric Spaces: sequences in metric spaces; complete metric spaces; examples of complete and non-complete metric spaces; sequence          characterization of a point belonging to the closure of a subset.
  5           Metric Spaces: continuity and uniform continuity of functions between metric spaces; examples of uniformly continuous functions;
               extension by uniform continuity.
  6           Metric Spaces: criterium for continuity of functions between metric spaces;  equivalent metrics.
  7           Metric Spaces: completion of a metric space; Cartesian product of finite number of metric spaces; convergence of sequences in
               cartesian product.
  8           Metric Spaces: convex functions; Holder and Minkowski inequalities.
  9           Compactness  in Metric Spaces: compact sets; compact metric spaces; sequential criteria of compactness; precompactness.
 10          Compactness  in Metric Spaces: Cartesian product of finite number of compact metric spaces; Heine - Borel theorem;
               compactness and continuity; homeomorphism theorem; continuous functions on compact subsets.
 11          Compactness in Metric Spaces: uniform convergence of sequences and series of functions; properties and examples.
 12          Connectedness in Metric Spaces: connected subsets; connected components; continuous functions and connectedness.
 13          Applications: contraction mapping theorems; Arzela-Ascoli theorem.
 14          Applications: Tietze extension theorem; Baire theorem.



Schedule:
 Tuesdays    08:40-10:30 M-13
 Thursdays  08:40-10:30 M-13

Grading: There will be two midterm exams (30%), and a final exam (40%).
Midterm 1: November 07, 2022, Monday at 17:40 (30%)
Midterm 2: December 12, 2022, Monday at 17:40 (30%)
Final Exam: to be announced During Final Weeks  (40%)

Make up for Exams:
One single make-up exam will given to students who have a legitimate excuse for missing one of the regular exams.

Academic Honesty:
The METU Honour Code is as follows: "Every member of METU community adopts the following honour code as one of the core principles of academic life and strives to develop an academic environment where continuous adherence to this code is promoted. The members of the METU community are reliable, responsible and honourable people who embrace only the success and recognition they deserve, and act with integrity in their use, evaluation and presentation of facts, data and documents."