Syllabus
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Course Code: MT-601 Course Name: (A) Real Analysis–II |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. | |
| 2 | Improper integrals and their convergence, Comparison tests, Abel’s and Dirichlet’s tests, Frullani’s integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter. | |
| 3 | Definition and examples of metric spaces, Open spheres and closed spheres, Neighbourhoods, Open sets, Interior, exterior and boundary points, Closed sets, Limit points and isolated points, Interior and closure of a set, Boundary of a set, Bounded sets, Distance between two sets, Diameter of a set, Subspace of a metric space. Cauchy and Convergent sequences, Completeness of metric spaces, Cantor’s intersection theorem. | |
| 4 | Dense sets and separable spaces, Nowhere dense sets and Baire’s category theorem, Continuous and uniformly continuous functions, Homeomorphism, Banach contraction principle. Compact spaces, Sequential compactness, Bolzano-Weierstrass property, Compactness and finite intersection property, Heine-Borel theorem.Continuous functions on compact spaces. |
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