Page 4 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

SYLLABUS

Measure Theory and Functional Analysis

Objectives: This course is designed for the of analysis of various types of spaces like Banach Spaces, Hilbert Space, etc. and also

Sr. No Description
1 Differentiation and Integration: Differentiation of monotone functions,

Functions of bounded variation,
2 Differentiation of an integral, Absolute continuity

3 Spaces, Holder, Minkowski inequalities, Convergence and Completeness

4 Bounded linear functional on the Lp spaces, Measure spaces, Measurable
Functions, Integration

5 General Convergence Theorems, Signed Measures, Radon-Nikodym

theorem.
6 Banach spaces: Definition and some examples, Continuous linear

transformations, The Hahn-Banach theorem

7 The natural imbedding of N in N**, The open mapping theorem, The
closed graph theorem,

8 The conjugate of an operator, The uniform boundedness theorem, The
uniform boundedness theorem, Hilbert spaces : The definition and some

simple properties

9 Orthogonal complements, Orthonormal Sets, The conjugate space H*, The
Adjoint of an Operator, Self Adjoint Operators

10 Normal and Unitary Operators, Projections, Finite dimensional spectral
theory : the spectrum of an operator on a finite dimensional

Hilbert space, the Spectral theorem