Duration: 21 weeks
Number of lectures per week: 3
Assessment: Regular assignments.
End-of-year Examination: One 3-hour examination
Description: Fundamental Concepts: Partial order, Zorn's lemma as an axiom, application to bases of vector spaces; cardinal numbers; ordinal numbers.
General Topology: Neighbourhoods, first countable, inadequacy of sequences, second- countable, (relationship to separability), continuity of functions at points, product topology (weak topology for continuous projections).
Nets, advantages over sequences, subnets; Hausdorff separation axiom, Urysohn's lemma, Tietze extension. Compactness via nets, Tychonoff's theorem (compactness of products), compactification (Stone-ech and universal properties, one-point), local compactness, completions of metric spaces, Baire category theorem.
Applications Fourier series in L2[0,2p], Wavelet bases for L2(R).
Objectives: This course aims to introduce general techniques used widely in analysis (and other branches of mathematics) and to treat a few topics that are active areas of research.
Jun 10, 1998