![]() ![]() Chapter 7 of Rudin’s book touches on a subtle but important issue in analysis: when can you interchange two limiting processes? For example, if a sequence of function f_n converges pointwise to a function f, is the limit of the integral (by definition a limit process) of f_n the same as the integral of the limit of f_n? This turned out to be a recurring theme in many later courses, and the study of which leads to many fascinating ideas such as uniform convergence and convergence theorems of Lebesgue integration. For instance, as soon as you write out definitions of continuity and closedness, it should be quite clear that “the zero set of a continuous real function is closed”. In the first half of the class, most of the proofs more or less “write themselves” once you have unraveled the definitions, so the most important part is again trying to understand what the statement is really saying. A general difficulty in studying this class is about writing proofs. It might be beneficial to go through every definition a few times to fully internalize the ideas. The second chapter on point-set topology is especially crucial in the sense that it lays foundation for the rest of the course as well as for any future analysis classes. ![]() However, the emphasis of the course is on the rigor of proofs. ![]() Many of the concepts such as differentiation and integration might already be familiar to students. This introductory class covers the first eight chapters of Walter Rudin’s “Principles of Mathematical Analysis”. MAT 215: Honors Analysis (Single Analysis) On this page, every non-introductory course except those numbered MAT 33x should count towards the real analysis departmental. In order to graduate with a mathematics degree, it is required to complete at least one real analysis course and one complex analysis course. The department also offers courses on the applications of analysis to other fields, including MAT 493/PHY 403 (Mathematical Methods of Physics). After taking the introductory courses, students interested in analysis often proceed to the four core analysis courses called the “Stein sequence,” described below. Analysis has applications ranging from physics to number theory, and underlies many branches of applied math. Princeton’s emphasis on analysis is reflected by the fact that two of the three introductory courses for math majors (MAT215 and MAT216) deal with the subject. Analysis is the study of various concepts that involve the idea of taking limits, such as differentiation, integration, and notions of convergence. ![]()
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