This course introduces principles of real analysis and the modern treatment of functions of one and several variables. Topics include metric spaces, the Heine-Borel theorem in R-n, Lebesgue measure, measurable functions, Lebesgue and Stieltjes integrals, Fubini's theorem, abstract integration, L-p classes, metric and Banach space properties, and Hilbert space.
Prerequisites: MA 232, MA 441
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MA441 and MA442 are capstone courses for mathematics majors. The two courses supply basic preparation for either graduate school or for application of mathematics to problems of science and engineering.
There are a number of different approaches to this course. The exact content of the course is not as important as the development of mathematical maturity by the students. Accordingly course outcomes are based on Bloom’s taxonomy.
- Know: Recall definitions and statements of theorems.
- Comprehend: Be able explain and restate theorems and definitions in different contexts and as they apply to special cases.
- Apply: Use the theorems and techniques taught in the course to solve problems.
- Analyze: Recognize which theorems and definitions apply to various situations.
- Synthesize: Be able to construct proofs.
The outcomes are repeated for each of the three major divisions of the course, namely
- Series of functions;
- Functions of several variables;
- Stokes Theorem, or Lebesgue integration.
Thus there are altogether fifteen course outcomes.
Walter Rudin, Principles of Mathematical Analysis
John O’Connor, A First Analysis Course
Frank Morgan, Real Analysis