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-Cech 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, l_p, closed subspaces,c₀, 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.

• Applications Fourier series in L₂[0,2p], Wavelet bases for L²(R).

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 12, 2005