# Math 425 - Fundamental Concepts of Analysis II

See UW General Catalog for course description and prerequisite information.

#### Textbook

• 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