Syllabus For The Subject FUNCTIONAL ANALYSIS

UNIT - 01

BANACH SPACES - DEFINITION AND EXAMPLES - HOLDER`S AND MINKOWSKI`S

INEQUALITIES (*) CONTINUOUS LINEAR TRANSFORMATIONS - EQUIVALENCE OF

VARIOUS NORMS IN 1NP.

UNIT - 02

LOCALLY COMPACT NORMAL LINEAR SPACE IN FINITE SEPARABLE IF N* IS SOCONJUGATE

SPACES 1NP AN 1N ? (*) NATURAL IMBEDDING OF N INTO N** ANY FINITE

DIMENSIONAL NORMAL LINEAR SPACE IS REFLEXIVE.

UNIT - 03

THE OPEN MAPPING THEOREM - THE CLOSED GRAPH THEOREM - CONJUGATE OF AN

OPERATOR.

UNIT - 04

HILBERT SPACES - SOME EXAMPLES - ORTHOGONAL COMPLEMENTS - ORTHONORMAL

RESULTS - A HILLERT SPACE H IS SEPARABLE IF AND ONLY IF EVERY ORTHONORMAL

SET IS COUNTABLE (*) ORTHOGONAL DIMENSION OF H (*).

UNIT - 05

THE CONJUGATE SPACE H* AD JOINT OF AN OPERATOR - SELF AD JOINT OPERATORS -

NORMAL AND UNITARY OPERATOR PROJECTIONS.

UNIT - 06

FNITE DIMENSIONAL SPECTRAL THEORY - MATRIUS - DETERMINANTS AND THE

SPECTRUM OF AN OPERATOR - THE SPECTRAL THEOREM

UNIT - 07

THE STRUCTURE OF COMMUTATIVE BONANCH ALGELUAS.

UNIT - 08

THE CRELFAND MAPPING APPLICATION OF THE FORMULAE R(X) = L1MLLXNLL1/N.

UNIT - 09

INVOLUTIONS IN BANACH ALGELUA.

UNIT - 10

THE GELFAND NEUMARK REPRESENTATIONS THEOREM.

Vinayaka Missions University,Directorate of Distance Education

Salem India

MASTER OF SCIENCE IN MATHEMATICS

2 Yr.

FUNCTIONAL ANALYSIS(2030511)