Math 425 - Fundamental Concepts of Analysis II

See UW General Catalog for course description and prerequisite information.


  • Principles of Mathematical Analysis (Third Edition) by Walter Rudin

Suggested Syllabus

Metric Spaces

  • Euclidean space, metric spaces
  • open and closed sets, interiors, closures
  • compactness, sequential compactness, Heine-Borel Theorem
  • connectedness, intervals
  • completeness of metric spaces and of the Euclidean space

Continuous Functions of Metric Spaces

  • epsilon-delta and sequence definitions of limits, properties
  • continuity of functions between metric spaces
  • continuity and connectedness
  • continuity and compactness
  • Extreme Value Theorem
  • uniform continuity of continuous maps of compact spaces

Functions of Several Variables

  • Normed linear spaces, completeness of C(X)
  • L(Rn,Rm) with the sup norm
  • continuity of the inversion map
  • definition of the derivative and necessity of continuity
  • chain rule, partial derivatives and matrix representation of the derivative
  • characterization of C1 functions
  • Mean Value Theorem
  • Inverse Function Theorem
  • Contraction Mapping Theorem
  • Implicit Function Theorem

Instructors Only