**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.

*Functional Analysis*:

**Banach spaces:**-
definitions and examples (C
_{0}(X),*l*_{¥}, C(K), Hölder and Minkowski inequalities,*l*_{p}, closed subspaces,c_{0}, L_{p}(**R**), L_{p}[0,1]). **Linear operators:**-
examples of continuous inclusions among
*l*_{p}and L_{p}[0,1] spaces, n-dimensional normed spaces isomorphic. Open mapping and closed graph theorems. Uniform boundedness principle. **Dual spaces:**- Hahn-Banach theorem, canonical isometric embedding in
double dual, reflexivity.
**Hilbert space:**- orthonormal bases (existence, countable if and only if
separable), orthogonal complements, Hilbert space direct sums,
bounded linear operators on a Hilbert space as a C
^{*}-algebra. Completely bounded and completely positive operators.

*Applications* Fourier series in L_{2}[0,2p], Wavelet bases for
L^{2}(**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