Speakers:

(i) Amin Sofi, Kashmir University, Srinagar
(ii) Vinayak Sholapurkar, Sir Parashurambhau College, Pune
(iii)Saurav Pal, Indian Institute of Technology, Powai, Mumbai
(iv) Diganta Borah, Indian Institute of Science Education and Research, Pune

Course Associates:
(i) KrishnatMasalkar, AbasahebGarware College, Pune
(ii) GeetanjaliPhatak, Sir Parashurambhau College, Pune

Syllabus:

Module 1. Introduction to Banach Spaces, Hahn-Banach theorems, Consequences of Hanh-Banach theorems, Convergence in L(X,Y). Uniform Boundedness Principle.
Module 2.Introduction to Hilbert Spaces, Bessel’s inequality, Complete orthonormal sets, Parseval’s identity, Complete orthonormal basis in L₂(0, 2),Riesz Representation theorem, weak convergence,Adjoint and sesquilinearfunctionals, compact normal operator
Module 3. Orthogonal Projections and Positive definite operators, square root of a positive operator, spectral decomposition, spectral theorem for compact normal operators.
Module 4. Properties of compact operators, spectral notions, spectrum of a compact operator, Self adjoint operators, spectral properties of selfadjoint operators, spectral theorem for self adjoint operators.

 Abstract of the Lectures:

(i) Dr. Amin Sofi: Introduction to Banach Spaces, Hahn-Banach theorems, Consequences of Hanh-Banach theorems, Convergence in L(X, Y). Uniform Boundedness Principle. The lectures will cover Module 1.

(ii) Dr.Diganta Borah: Introduction to Hilbert Spaces, Bassel’s inequality, Complete orthonormal sets, Parseval’s identity, Complete orthonormal basis in L_2(0, 2\pi). Riesz Representation theorem, weak convergence. Adjoint and sesquilinearfunctionals, compact normal operators. The lectures will cover Module 2.

(iii) Dr.Saurav Pal: Orthogonal Projections and Positive definite operators, square root of a positive operator, spectral decomposition, spectral theorem for compact normal operators. The lectures will cover Module 3.

(iv) Dr.V. M. Sholapurkar:Properties of compact operators, spectral notions, spectrum of a compact operator. Self adjoint operators, spectral properties of self adjoint operators, spectral theorem for self adjoint operators. The lectures will cover  Module 4.

References:
1. Bachman and Narici, Functional Analysis, Academic Press, 1966
2. B. V. Limaye, Functional Analysis, New Age International, third edition, 2017

Time Table:
First Week: 23rd October to 04th November, 2017

Second Week: 23^rd to 28^th November, 2015

SP: Saurav Pal    
AS: Amin Sofi        
VS: VinayakSholapurkar    
DB:Diganta Borah
KM:KrishnatMasalkar    
GP: Geetanjali Phatak