MA607

Linear Operators and Generalized Functions (4-0-0) 4

Basic topological notions in a metric space. Continuity, Convergence, Cauchy sequences and completeness.Contraction mapping theorem. 

Hilbert spaces. Projection theorem. Separable Hilbert spaces and orthonormal bases. Linear functionals. Riesz representation theorem.

Elements of Lebesgue integration: Measurable sets and functions. Integral of a nonnegative measurable function. MCT. Fatou’s lemma. Integral functions. Properties of the integral. LDCT. L^P - spaces. Product measures and Fubini’s theorem. 

Operator theory in Hilbert spaces. Bounded linear operators. Adjoint of an operator. Solvability conditions. Spectrum of an operator. Compact operators. Fredholm alternative. Eigen function expansions.

Distributions and Fourier transforms. Test functions. Definition of a distribution. Calculus of distributions. Operations on distributions. Schwartz class S. Fourier transform on S. Fourier transform of L² – functions. Tempered distributions and their Fourier transforms.

References: 

BARTLE, R. G., Elements of Integration, Wiley.

GAQUET, C. and WITOMSKI, P., Fourier Analysis and Applications, Springer.

STAKGOLD, I., Green’s Functions and Boundary value Problems (e), Wiley.

STRICHARTZ, R., A Guide to Distribution Theory and Fourier Transforms,World Scientific.