## Details of MA4102 (Autumn 2019)

Level: 4 |
Type: Theory |
Credits: 4.0 |

Course Code | Course Name | Instructor(s) |
---|---|---|

MA4102 |
Functional Analysis |
Shibananda Biswas |

Syllabus |
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MA4102 : Functional Analysis
Normed Linear Spaces: Definitions, Banach Spaces, Hilbert spaces, non-compactness of the unit ball in infinite dimensional normed linear spaces, quotient spaces. Linear Maps: Boundedness and continuity, linear functionals. isometries, Mazur-Ulam theorem on isometries. Completeness: Banach-Steinhaus theorem, open mapping theorem and closed graph theorem. Convexity: Hahn-Banach extension theorem, complex Hahn-Banach theorem, separation of convex sets, applications. Duality: Dual spaces, Riesz representation theorem, reflexivity, Eberlain-Schmulian theorem, weak topologies, weak convergence, weak compactness, Banach-Alaoglu theorem, Krein-Milman theorem, adjoints and compact operators. Hilbert Spaces: Bessels inequality, complete systems, Gram-Schmidt orthogonalization, Parsevals identity, projections, orthogonal decomposition, bounded linear functionals in Hilbert spaces. Spectral Theory: Spectrum, Fredholm theory of compact operators, spectral theory of compact self-adjoint operators, minimax principle, application to integral operators. |

Prerequisite |
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Analysis IV (MA3204) and Topology (MA3201). |

References |
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Suggested Texts/Reference Books:
1. Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer. 2. Bollabs, B., Linear Analysis: An Introductory Course, Cambridge University Press. 3. Conway, J.B., A Course in Functional Analysis (2nd Edition), Springer-Verlag. 4. Eidelman, Y., Milman, V. and Tsolomitis, A., Functional Analysis: An Introduction,American Mathematical Society. 5. Kesavan, S., Functional Analysis, Hindustan Book Agency. 6. Lax, P.D., Functional Analysis, Wiley-Interscience. 7. Limaye, B.V., Functional Analysis, New Age Publishers. 8. Rudin, W., Functional Analysis, McGraw-Hill. 9. Simmons, G.F., Topology and Modern Analysis, Tata McGraw-Hill. |

#### Course Credit Options

Sl. No. | Programme | Semester No | Course Choice |
---|---|---|---|

1 | IP | 1 | Not Allowed |

2 | IP | 3 | Core |

3 | IP | 5 | Not Allowed |

4 | MR | 1 | Not Allowed |

5 | MR | 3 | Not Allowed |

6 | MS | 7 | Core |

7 | RS | 1 | Elective |

8 | RS | 2 | Elective |