Banach spaces – The definition and some examples – Continuous linear Transformations – The Hahn- Banach theorem – The natural imbedding of N in N** - The open mapping problem.
The conjugate of an operator – Hilbert spaces – The definition and some simple properties – Orthogonal complements - Orthonormal sets.
The Conjugate space H* - The adjoint of an operator – Self-adjoint operators – Normal and unitary operators – Projections.
Matrices – Determinants and the spectrum of an operator – The spectral theorem.
The definition and some examples of Banach Algebra – Regular and singular elements – Topological divisors of zero – The spectrum – The formula for the spectral radius.
Reference Book:
1. Foundations of Functional Analysis by S.Ponnusamy, Narosa, Seventh Reprint, 2012. 2. A First Course in Functional Analysis by D.Somasundaram, Second Reprint, 2012, Narosa Publishers. 3. Functional Analysis by Kosaku Yosida, Second Indian Reprint 2008, Springer Verlag.
Text Book:
Introduction to Topology and Modern Analysis by G.F. Simmons, McGraw – Hill Book Company, London, 2011. Unit I: Chapter 9: Sections: 46 – 50. Unit II: Chapter 9: Sections: 51, Chapter 10: Section: 52-54. Unit III: Chapter 10: Sections: 55 – 59. Unit IV: Chapter 11: Sections: 60 – 62. Unit V: Chapter 12: Sections: 64 – 68.
