SWIM 2026
Real Analysis
This lecture series provides a comprehensive introduction to real analysis, drawing on Terence Tao's Analysis I. Structured to build from foundational concepts to advanced theorems, it begins with the Axioms of Completeness and Least Upper Bound Property, exploring real number structure, sequences (convergence, limits, Cauchy sequences), and key results like Bolzano-Weierstrass. The series transitions to continuity (epsilon-delta definitions, Intermediate/Extreme Value Theorems), uniform continuity, and algebraic properties of continuous functions. Later lectures focus on differentiation (derivatives, Mean Value Theorem) and applications including Inverse Function Theorem and L’Hôpital’s Rule. Emphasizing rigorous proofs and central theorems, the series connects theoretical foundations with analytical tools, offering systematic progression from real numbers to function behavior while highlighting Tao's pedagogical approach.
Lecture Schedule & Resources
Note: The schedule is tentative and may be subject to change during the program.
| Lecture | Topics Covered | Speaker | Resources |
|---|---|---|---|
| Lecture 1 May 11 (Mon) |
Introduction to Sequences: Definition of a sequence, convergence, limits, uniqueness of limits, bounded sequences. | B V Rao – Archana M P | |
| Lecture 2 May 12 (Tue) |
Limit Laws, Monotonicity, and Cauchy Sequences: Limit laws, monotonic sequences (definition, examples), Cauchy sequences, completeness via Cauchy. | B V Rao – Archana M P | |
| Lecture 3 May 14 (Thu) |
Subsequences and Bolzano-Weierstrass: Subsequences, convergence of subsequences, Bolzano-Weierstrass Theorem. | B V Rao – Archana M P | |
| Lecture 4 May 15 (Fri) |
Defining Continuity: The epsilon-delta definition of continuity, the sequential criterion for continuity. | B V Rao – Archana M P |
Notes |
| Lecture 5 May 18 (Mon) |
Examples and Counterexamples of Continuity: Detailed analysis of continuity in functions (e.g., polynomials, piecewise functions), common discontinuities. | K. S. Senthil Rani – Arindam Sutradhar | |
| Lecture 6 May 19 (Tue) |
Intermediate and Extreme Value Theorems: The Intermediate Value Theorem for continuous functions, the Extreme Value Theorem on closed intervals. | K. S. Senthil Rani – Arindam Sutradhar | |
| Lecture 7 May 20 (Wed) |
Algebra of Continuous Functions: Continuity of sums, differences, products, quotients, and compositions of continuous functions. | K. S. Senthil Rani – Arindam Sutradhar | |
| Lecture 8 May 22 (Fri) |
Uniform Continuity and Compactness: Definition of uniform continuity, contrast with pointwise continuity, uniform continuity on compact sets. | K. S. Senthil Rani – Arindam Sutradhar | |
| Lecture 9 May 25 (Mon) |
Introduction to Differentiation: Definition of the derivative, differentiability implies continuity, geometric interpretation of the derivative. | Anwoy Maitra – Praveena P | |
| Lecture 10 May 26 (Tue) |
Differentiation Rules and Local Extrema: Basic differentiation rules (sum, product, quotient, chain rule), definition of local maxima and minima, Fermat's Theorem. | Anwoy Maitra – Praveena P | |
| Lecture 11 May 28 (Thu) |
Mean Value Theorem and Applications: Rolle's Theorem, the Mean Value Theorem, and applications including using the derivative to determine monotonicity. | Anwoy Maitra – Praveena P | |
| Lecture 12 May 29 (Fri) |
Inverse Function Theorem and L'Hôpital's Rule: Differentiability of inverse functions and the Inverse Function Theorem, L'Hôpital's Rule for evaluating indeterminate forms. | Anwoy Maitra – Praveena P |