TEW  Functional Analysis (2021)
Speakers and Syllabus
Name of the Speaker with affiliation 
No. of Lectures 
Detailed Syllabus 
1.Prof. Thamban Nair, Prof. Pradipta Bandopadhyay. 
9 hours 
A quick Introduction to Banach spaces and bounded linear maps, Uniform boundedness principle (with some applications to classical analysis), Open mapping theorem (Closed graph theorem), Hilbert space (orthonormal systems, orthogonal basis, bounded linear maps), Gram Schmidt orthogonalisation process (applied to certain concrete sequences in spaces), Projection Theorem, Riesz representation theorem for Hilbert space 
2.Prof. Sameer Chavan, 
5 hours 
Unitary operators and the unitary group on Hilbert space, Spectral theorem self adjoint operators, Hilbert space methods in function theory (including Maximum modulus theorem, MuntzSzas theorem for

3.Dr. Surinderpal Singh 
6 hours 
ArzelaAscoli theorem, BanachMazurkiewicz theorem on the size of C\ND. Hausdorff Alexandroff theorem: (Each compact metrics space as a quotient of the Cantor set), Banach Mazur theorem (C[0,1] as a universal separable Banach space), Computation of the dual of C(K) for K = [0,1] and K = compact subset of ℝ as a prelude to the Riesz Representation Theorem for C(K), K a compact metric space.

4.Prof. M. A. Sofi 
7 hours 
Background material and motivation, Weak, weak* topologies on Banach spaces, Banach  Alaoglu theorem and applications, Analytic and geometric forms of the HBT and their equivalence, Applications of HBT, Invariant form of HBtheorem, Construction of universal, finitely additive invariant measures on ℝ and . 
References:

(Prof. Sameer Chavan)
(i) Operator Analysis: Hilbert Space Methods in Complex Analysis, J. Agler, J. E. McCarthy,N. J. Young, Cambridge University Press,2020.
(ii) Simon, Operator Theory. A Comprehensive Course in Analysis, Parts I andV, American Mathematical Society, Providence, RI, 2015,
(iii) J.B. Conway, A course in Functional Analysis, Springer, 2006.

(Prof. Pradipta Bandopadhyay)

S. Kumaresan and D. Sukumar, Functional Analysis, Narosa.

R. Bhatia, Notes on Functional Analysis, Hindustan Book Agency

M. Fabian et al, Banach Space Theory, Springer Verlag

Dr. Surinder Pal Singh Kainth)
(a) Arzela Ascoli theorem (reference (iv)).
(b) Hausdorff–Alexandroff Theorem (Proposition 2.3, reference (ii))
(c) Riesz Representation Theorem for C(K) when K = [0,1] or a compact subset of ℝ (reference (i)).
(d) Riesz Representation Theorem for C(K) when K is a compact metric space (reference (ii))
(e) BanachMazur Theorem. (page 18 of reference (iii)).
References:
 Rafael del Rio, A. L. Franco, and J. A. Lara; An approach to F. Riesz representation Theorem, CUBO A Mathematical Journal, 20 (2), 2018, 112.
 V. S. Sunder; The Riesz Representation Theorem, Indian Journal of Pure and Applied Mathematics, 39 (6), 2008, 467481.
 F. Albiac, N. J. Kalton; Topics in Banach space theory, Springer, 2006
 W. Rudin; Principles of mathematical analysis, McGrawHill, 1976.

Prof. M. A. Sofi

M. Fabian et al, Banach Space Theory, Springer Verlag

Some Problems in functional analysis inspired by Hahn Banach theorem, Annals of Functional Analysis, 5(2014) no.2, 129.

Weak* topology, compactness and the existence of certain invariant measures, Mathematics Newsletter (Ramanujan Math Soc), 31(2), 4961, Sept. 2020.
Tutors:
S. No. 
Name 
Affiliation 
1 
Dr. Ramiz Reza/Dr. Nisar A. Lone 
IIT Jammu/JKIMS, Srinagar 
2 
Dr. Ajay Kumar 
IIT Jammu 
Guest Speaker:
S. No. 
Name 
Affiliation 
1 
Prof. Thamban Nair 
Department of Mathematics, IIT Madras 
Note: All the speakers shall be expected to provide notes of their lectures.
Time Table
TimetableI (Sundays):
Day 
Date 
Lecture1 (1011.30) 
Tea (11.3511.55) 
Lecturer 2 (12.00–1.00) 
Lunch (1.30 – 3.00) 
Lecture 3 (3.004.00) 
Discussion (4.155:15) 
Tea & Snacks 


(Speaker’s name) 

(Speaker’s name) 

(Speaker’s name) 
(Tutor’s name) 

Sun 
Nov.28 
TN 
Tea 
TN 
Lunch 
SC 
Discussion (TN+ 

Sun 
Decc.5 
MAS 
Tea 
TN 
Lunch 
Guest Lecture 
Discussion (SC+ 
Snacks 
Sun 
Dec.12 
Discussion (SPS) 
Tea 
MAS 
Lunch 
Panel Discussion on Teaching and Research in UGMaths (All the speakers + participants) 
Snacks 
Time TableII (Weekdays)
Day 
Date 
Lecture /Discussion 
Mon 
Nov.29 
SC 
Tue 
Nov.30 
SC 
Wed 
Dec.1 
PB 
Thurs 
Dec.2 
PB 
Fri 
Dec.3 
PB 
Sat 
Dec.4 
MAS 
Mon 
Dec.6 
MAS 
Tue 
Dec.7 
Discussion (MAS) 
Wed 
Dec.8 
SPS 
Thurs 
Dec.9 
SPS 
Fri 
Dec.10 
SPS 
Sat 
Dec.11 
Guest Lecture (TN) 
Speakers
TN: Prof. Thamban Nair (IIT Madras)
PB: Prof. Pradipta Bandopadhyay (ISI, Kolkatta)
SC: Prof. Sameer Chavan (IIT Kanpur)
SPS: Dr. Surinder Pal Singh Kainth (Panjab University, Chandigarh)
MAS: Prof. M A Sofi, JKIMS (Kashmir University), Srinagar