**Requirements/prerequisites:** The material in 221 is going to be relied upon.

**Duration:** 19 weeks

**Number of lectures per week:** 3

**Assessment:** Regular assignments.

**End-of-year Examination:** One 3-hour examination

**Description: **

The following is a draft syllabus.

**Banach spaces:**-
definitions and examples (C(X),
*l*^{p}, Hölder and Minkowski inequalities, closed subspaces, c_{0}, L^{p}(**R**), L^{p}[0,1]). **Completeness for metric spaces:**- completion.
Baire category.
**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. **Fundamental Concepts:**-
Partial order, Zorn's lemma as an axiom, application to bases
of vector spaces; cardinal numbers; ordinal numbers.
**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].

There is a web site `http://www.maths.tcd.ie/~richardt/321`
for the course.

**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.

Oct 5, 2008

File translated from T

On 5 Oct 2008, 17:47.