ANDHRA UNIVERSITY, WALTAIR

 

M.Sc. APPLIED MATHEMATICS, FOURTH SEMSTER

 

AM-401 : FUNCTIONAL ANALYSIS

 

[ A total of seven questions are to be set.  All questions carry equal marks.  The first question which is compulsory carries 20 marks.  It consists of  4 short answer sub questions each of five marks covering the entire syllabus.  The remaining six questions each carrying 20 marks are to be set as suggested in the body of the syllabi.]

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Algebraic Systems: Groups, rings, structure of rings, Linear spaces, dimension of linear space, linear transformations, algebras.

Branch  spaces:   The definition and some examples, continuous linear transformations, The Hahn-Banach theorem, The natural imbedding of N in N^** , The open mapping theorem, The conjugate of an operator.  (Chapters 8 and 9 of text book).

[Three questions are to be set]

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.

Finite – Dimensional spectral theory: Matrices, Determinants and the spectrum of an operator, the spectral theorem, a survey of the situation.  (chapters 10 and 11 of text book)

[Three questions are to be set]

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Text book:

 G.F. Simons- Mc Graw Hill ‘Introduction to Topology and Modern Analyis.