Course Content: 

  1. Normed linear spaces, Banach spaces, Equivalence of norms on finite dimensional spaces, Riesz lemma and  characterization of   finite dimensional normed spaces,  Hamel basis and   Schauder basis of normed spaces, Separable normed spaces. (10 hours)
  2. Bounded linear operators, Continuous  linear functionals, Hahn-Banach theorems (separation and extension theorems), Dual spaces and transpose, Dual of  some classical spaces like c0, lp, Lp(for p≥ 1); Bidual of a normed linear space and Reflexive spaces. (12 hours)
  3. Uniform boundedness principle, Open mapping theorem, Closed graph theorem.(5 hours)
  4. Compact operators, Spectra of bounded linear operators and compact operators. (7 hours)
  5. Hilbert spaces, Bessel’s inequality, Orthonormal basis, Separable Hilbert space, Projection theorem and  Riesz Representation Theorem. (8 hours)

 

Learning Outcomes:  Upon completing the course, students will be able: 

  • To learn to recognize the fundamental properties of normed spaces and of the operators between them.
  • To understand and apply fundamental theorems from the theory of normed  spaces, including the Hahn-Banach theorem, the open mapping theorem and uniform boundedness principle. 
  • To understand the concepts of spectra of operators.
  • To apply Functional Analysis techniques to problems arising in Partial Differential Equations, wavelet analysis and other branches of Mathematics.


Text books:

  1. J. B. Conway, A course in functional analysis, GTM (96), Springer (Indian reprint 2006).
  2. B. V. Limaye, Functional Analysis, 2nd Ed., New Age International Publishers, 1996. 

References :  

 

  1. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, 2nd Ed., Academic Press, 1980.
  2. G. Bachman and L. Narici,  Functional Analysis, Dover Publication, 2012
  3. S. Kesavan, Functional Analysis, TRIM series, Hindustan Book Agency, 2009 
  4. Yosida, K., Functional Analysis (4th Edition), Springer-Verlag Berlin Heidelberg, 1995
  5. Rajendra Bhatia, Notes on Functional Analysis, Hindustan Book Agency, 2015
  6. M. T. Nair, Functional analysis, PHI-Learning, New Delhi, Fourth Print 2014