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)
- 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