### 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:

1. Tao, T. 2014., Analysis I, 3rd edition, Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency.
2. Tao, T. 2014., Analysis II, 3rd edition, Texts and Readings in Mathematics, vol. 38, Hindustan Book Agency.
3. Apostol, T. M., Mathematical Analysis, 2nd edition, Narosa.

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Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 20 Feb 2020