Instructor: Dr. András STIPSICZ

Text: handouts, and R. G. Bartle, The elements of real analysis
 

Prerequisite: Calculus

Course description: We cover introductory material in analysis, with complete proofs and detailed explanation of meaning and
importance of notions introduced. The course ends with multivariable calculus.

Topics:

1.  Metric spaces, topological spaces
 

• Basic definitions and examples (standard metrics, discrete metric spaces)
• Sequences (convergent and Cauchy sequences, Bolzano-Weierstrass selection principle, complete metric spaces)
• Basic concepts of topology (open and closed sets, compact sets, connectedness)
• Topology of metric spaces (Heine-Borel theorem, Banach fixed point theorem)
• Continuous functions (intermediate value theorem, min-max. value theorem)

2. Infinite series

• Convergent series (absolute and conditional convergence)
• Tests for convergence (comparison, limit comparison, root, ratio, alternating series test)
• Power series (convergence radius, Cauchy-Hadamard formula)

• Limit of functions and differentiation (definitions and basis properties, mean value theorem, fundamental theorem of calculus)
• Differentiation in  Rⁿ (Implicit and Inverse function theorems)