School of Mathematics
Course 321 - Modern Analysis 1998-99 (Optional JS & SS Mathematics, SS Two-subject Moderatorship )
Lecturer: Dr. R.M. Timoney
Requirements/prerequisites: 212

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 (C0(X), l, C(K), Hölder and Minkowski inequalities, lp, closed subspaces,c0, Lp(R), Lp[0,1]).

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.

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], 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