# Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper 1- Functional Analysis-II Syllabus

## Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper 1- Functional Analysis-II Syllabus

Theory Paper: Max. Marks- 42

Internal Assessment: Max. Marks- 08

Note: The question paper will consist of three sections A, B&C. Section A will consist of 7 objective ty pe questions each carrying I marks, section B will consist of 5 short answer type questions each carrying 3 marks and section C will consist of 2 long answer type questions each carrying 10 marks. Section B& C will have internal choice.

Unit 1- Uniform boundedness principle and some of its consequences, Open mapping and closed graph theorems.

Unit 2- Hahn-Banach theorem for real linear spaces, Hahn-Banach theorem for complex linear spaces and normed linear spaces and some of its consequences, Reflexivity of Normed spaces.

Unit 3- Inner product spaces, Examples and Properties, Convex sets, Riesz lemma on closed convex set, orthogonality of vectors, Projection theorem, Hilbert spaces.

Unit 4- Orthonormal Sets, Bessel's inequality, Complete orthonormal sets and Parseval's Identity, Riesz representation theorem, Reflexivity of Hilbert spaces.

Unit 5- Adjoint of an operator on a Hilbert space, Self-adjoint operators, Positive operators, Projection, Normal and Unitary operators.

Recommended Books:

[1]. P.K.Jain, 0.P. Ahuja and Khalil Ahmad, Functional Analysis,

[2]. K.K.Jha, Functional Analysis,

[3]. B.V. Limage, Functional Analysis,

[4]. G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw IlI, New York.

[5]. S.K. Boose, Functional Analysis,

Tags