FUNCTIONAL ANALYSIS- I (Compulsory Paper)
- Cover theoretical needs of Partial Differential Equations and Mathematical Analysis.
- Inter-relate the problems arising in Partial Differential Equations, Measure Theory and other branches of Mathematics.
- Know about various spaces such as Normed Linear Spaces, Banach spaces.
- Use the operators on these spaces.
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Learning Outcomes |
Learning and teaching strategies |
Assessment |
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After the completion of the course the students will be able to: CLO54- Explain the fundamental concepts of functional analysis in applied contexts. CLO55- Use elementary properties of Banach space and Hilbert space. CLO56- Identify normal, self adjoint or unitary operators. CLO57- Communicate the spectrum of bounded linear operator. CLO58- Construct orthonormal sets.
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Approach in teaching: Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching Learning activities for the students: Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical
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Presentations by Individual Students Class Tests at Periodic Intervals. Written assignment(s) Semester End Examination |
Normed linear spaces, Quotient space of normed linear spaces and its completeness, Banach spaces and examples, Bounded linear transformations.
Normed linear space of bounded linear transformations, Equivalent norms, Basic properties of finite dimensional normed linear spaces and compactness, Riesz lemma.
Open mapping theorem, Closed graph theorem, Uniform boundness theorem, Continuous linear functional, Hahn-Banach theorem and its consequences.
Hilbert space and its properties,Orthogonality and functionals in Hilbert spaces, Phythagorean theorem, Projection theorem, Orthonormal sets.
Bessel’s inequality, Complete orthonormal sets, Parseval’s identity, Structure of a Hilbert space, Riesz representation theorem.
- G.F.Simmons,Topology and Modern Analysis, Mc-Graw Hill, 1963.
- G.Bachman, Lawrence Narici,Functional Analysis, Academic Press, 1966.
- Dileep S.Chauhan, Functional Analysis and calculus in Banach space, Jaipur Publishing House, 2013
- B.V. Limaye, Functional Analysis, New Age International, New Delhi,2017.
- Erwin Kreyszig, Introductory Functional Analysis with Application, Willey, 2007.
- A.E.Taylor,Introduction to Functional analysis, John Wiley and Sons, 1958.
- Graham Allan, H. Garth Dales, Introduction to Banach Spaces and Algebras, Oxford University Press, 2010.
- Reinhold Meise, Dietmar Vogt, M. S. Ramanujan, Introduction to Functional analysis,Oxford University Press,1997.
- A.L.Brown, A. Page, Elements of Functional Analysis, Van Nostrand Reinhold, 1970.
- F. Riesz, B. Sz. Nagay, Functional Analysis, Dover Publications, 1965.
