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, c₀, 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 L²[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