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Every Spring and Fall semester, some Summers
Real numbers, topology of Euclidean spaces, Cauchy sequences, completeness, continuity and compactness, uniform continuity, series of functions, Fourier series
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)
- 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