School of Mathematics School of Mathematics
Course 321 - Functional Analysis 2008-09 (Optional JS & SS Mathematics, SS Two-subject Moderatorship )
Lecturer: Prof. R. Timoney

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


The following is a draft syllabus.

Banach spaces:
definitions and examples (C(X), lp, Hölder and Minkowski inequalities, closed subspaces, c0, Lp(R), Lp[0,1]).

Completeness for metric spaces:
completion. Baire category.

Linear operators:
examples of continuous inclusions among lp and Lp[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.

Fourier series in L2[0,2p].

There is a web site 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 TEX by TTH, version 2.70.
On 5 Oct 2008, 17:47.