## Details of MA4102 (Autumn 2018)

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

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

MA4102 |
Functional Analysis |
Saugata Bandyopadhyay |

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. |

References |
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Suggested Texts/Reference Books:
1. Bollabs, B., Linear Analysis: An Introductory Course, Cambridge University Press, 1990. 2. Conway, J. B., A Course in Functional Analysis (2nd Edition), Graduate Texts in Mathematics, Vol 96, Springer-Verlag, Berlin, 1990. 3. Eidelman, Y., Milman, V. and Tsolomitis, A., Functional Analysis : An Introduction, Graduate Studies in Mathematics, Vol 66, American Mathematical Society, 2004. 4. Kreyszig, E., Introductory Functional Analysis with Applications, Wiley, 1978. 5. Lax, P. D., Functional Analysis, Wiley, 2002. 6. Rudin, W., Functional Analysis (2nd Edition), McGraw-Hill, 1991. |

#### 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 |