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522 - Analysis II

Prerequisites: 
Math 521, also a course in linear algebra which can be taken concurrently.
Frequency: 
Fall (I), Spring (II)
Student Body: 

Math majors, Physics and Engineering majors and graduate students in related areas

Credits: 
3. (N-A)
Recent Texts: 
Principles of Mathematical Analysis, by W. Rudin. Mathematical Analysis, by A. Browder.
Course Coordinator: 
Andreas Seeger
Background and Goals: 

The sequence Math 521-522-621 has the objective of conveying an understanding of the structure of analysis in itself as well as its role as a tool for other disciplines. This sequence is highly recommended for any math major and it is essential for students preparing for graduate studies in mathematics; also it should be taken by students in physics and engineering who intend to do graduate work in their areas.

Alternatives: 
N/A
Subsequent Courses: 
Math 621, also Math 623 and 629.
Course Content: 
  • More on convergence.

                        Review of uniform convergence
 
                       Exponential functions, and more on power series.
                       Approximations of the identity.
                       Fourier series.
                       Approximation by polynomials, the Stone-Weierstrass theorem.
                       Infinite products.
                       Stirling's formula and the Gamma-function

  • Compactness in metric spaces.
    • Characterizations of compactness.
    • Arzela-Ascoli theorem.
    • Applications (such as Peano's existence theorem for differential equations).
  • The contraction principle.
    • With applications, in particular to differential equations.
  • Differential calculus.
    • Differentiability in normed spaces, the derivative as a linear map.
    • Chain rule.
    • Maps from R^m into R^n.
    • Taylor's formula.
    • Inverse mapping theorem.
    • Implicit functions.
  • Other topics (optional), such as:
    • Rectifiabilty of curves.
    • The construction of real numbers.
    • Baire category with applications.