UM 204: Introduction to Basic Analysis
Credits: 3:1
Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. The topology of $\R^n$: topology induced by norms, the Heine-Borel theorem, connected sets. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity. Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus. Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem.
Suggested books and references:
- Tao, T. 2014., Analysis I, 3rd edition, Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency.
- Tao, T. 2014., Analysis II, 3rd edition, Texts and Readings in Mathematics, vol. 38, Hindustan Book Agency.
- Apostol, T. M., Mathematical Analysis, 2nd edition, Narosa.