While the term “real analysis” initially referred to the theory of functions of a real variable, it has evolved to encompass a broader range of more general and abstract topics that form the foundation of modern analysis. This course will explore these fundamental theories and their applications.
We will cover a variety of topics, including:
1) Sigma-algebras and measures
2) Outer measures and Borel measures on the real line
3) Lebesgue measure on the real line and measurable functions
4) Integration of non-negative and complex functions
5) Modes of convergence
6) Finite product measures and the d-dimensional Lebesgue measure and integral
7) Integration in polar coordinates
8) Infinite product measures and signed measures
9) The Lebesgue–Radon–Nikodym theorem
10) Complex measures and differentiation in Euclidean space
11) Functions of bounded variation
12) Hilbert spaces and Lp spaces
Through this comprehensive curriculum, students will develop a deep understanding of both the theoretical underpinnings and practical applications of real analysis, essential for further study in mathematics and related fields.
“Real Analysis: Modern Techniques and Their Applications” by Gerald B. Folland
A solid understanding of the classical theory of functions of a real variable, including limits and continuity, differentiation and Riemann integration, infinite series, and uniform convergence is essential. Additionally, familiarity with the arithmetic of complex numbers and some elementary set theory and topology is required. A basic knowledge of linear algebra, focusing on the definitions of vector spaces, linear mappings, and determinants, is also expected. This foundational knowledge is crucial for success in the course.