Analysis I

Course Number: 
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Typical Scheduling: 
Every Spring and Fall semester

Real numbers, topology of Euclidean spaces, Cauchy sequences, completeness, continuity and compactness, uniform continuity, series of functions, Fourier series

Course Text: 

At the level of Elements of Real Analysis, Bartle (Sections 1-12, 14-18, 20-26, 34-38)

or Introduction to Analysis, Rosenlicht (Chapters 1, 2, 3, 4, Sections 7.2, 7.3)


Topic Outline: 
  • Upper bounds, least upper bounds, completeness of the real numbers.
  • Topology of Euclidean spaces: norms, the Cauchy-Schwarz inequality, open and closed sets, compactness, the Bolzano-Weierstrass and Heine-Borel theorems and connectedness.
  • Sequences in Euclidean spaces: Cauchy sequences, monotone sequences, pointwise, uniform and L2 convergence of sequences of functions
  • Continuity: Local properties of continuous functions. Preservation of compactness and connectedness. Uniform continuity. Contractions and operator norm continuity of linear operators
  • Sequences of continuous functions: Bernstein and Weierstrass approximation theorems
  • Infinite series: Convergence and absolute convergence, convergence tests
  • Series of functions. Power series, Fourier series