Syllabus

Topological spaces, methods of generating topologies, continuous mappings, axioms of separation, Frechet spaces, subspaces, Cartesian products, quotient spaces, function spaces, compact spaces, locally compact spaces, compactifications, countably compact spaces, pseudocompact spaces, sequentially compact spaces, totally bounded and complete metric spaces, paracompact spaces, countably paracompact spaces, connected spaces, kinds of disconnectedness, dimension of topological spaces and its basic properties, uniform spaces, totally bounded, complete and compact uniform spaces, proximity spaces.

Course Outline

Syllabus

Normes spaces, Banach spaces and Hilbert spaces, classical examples (sequence spaces and function spaces). Basic theorems.
General theory of topological vector spaces, locally convex spaces, separation theorems.
Weak topologies, theorems of Mazur, Alaoglu and Goldstine, weak compactness.
Schauder bases and basic sequences.
Extreme points, Krein Milman theorem.
Riesz representation theorem, Lp spaces.
Fixed point theorems.

Course Outline

Syllabus

Holomorphic, entire and meromorphic functions. Conformal mappings. Analytic continuations. Weierstrass' convergence theorem. The Gamma function. Infinite products. Riemann's mapping theorem. Harmonic functions and applications.

Course Outline

Syllabus

Second order o.d.e.’s: Sturmian theorems, Oscillation theorems. Differential inequalities and applications. Study by reduction of differential equations and problems to integral equations. Volterra and Fredholm differential equations. Resolvents, eigenvalues, eigenfunctions. Integral inequalities. Equations with distributed arguments. Delay equations and systems: solutions by the method of steps, existence and uniqueness by the use of fixed point theorems, stability. Fractional derivatives and fractional differential equations: basic notions and calculus, existence of solutions to initial value problems and boundary value problems. Time scales and dynamic equations : basic notions and calculus, existence of solutions to initial value problems and boundary value problems. Other topics.

Course Outline